New Graph

          [4, 4, 4, 7, 7, 7, 1, 1, 1], [2, 9, 5, 8, 3, 8, 5, 6, 2]

 

 

π = [3, 2, 1, 3, 2, 1, 3, 2, 1]

POSSIBLE RANKS

1 x 18
2 x 9
3 x 6

BASE DETERMINANT 2151937075/68719476736, .3131480589e-1

NullSpace of Δ

{3, 7, 8}, {1, 2, 4, 5, 6, 9}

Nullspace of A

[{3, 8},{7}] `,` [{2, 5, 6, 9},{1, 4}]

 

 
1 . Coloring, {}

R: [4, 4, 4, 7, 7, 7, 1, 1, 1]    B: [2, 9, 5, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `3` (` 1 + τ ` )`` (` - 3 + τ ` )` , 6` (` - 1 + τ ` )` , -3` (` - 1 + τ ` )` 2 , 3` (` 1 + τ ` )`` (` - 3 + τ ` )` , 6` (` - 1 + τ ` )` , -3` (` - 1 + τ ` )` 2 , 3` (` 1 + τ ` )`` (` - 3 + τ ` )` , 6` (` - 1 + τ ` )` , -3` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-15, -4, -1, -15, -4, -1, -15, -4, -1] . FixedPtCheck, [15, 4, 1, 15, 4, 1, 15, 4, 1]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y1 - y2, y1, y2, -y1 - y2, y1, y2, -y1 - y2, y1, y2]
  p' = s 5 - 2s 6   p' = s 4 - 4s 6   p' = s 3 - 8s 6   p' = s 2 - 16s 6   p = s 2 - 32s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 7 2 vs 7 2 vs 7 1 vs 3 2 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 1 .    order:   3

See Matrix
 

[y1, 0, 0, y1, 0, 0, y1, 0, 0]

 

  p = - s + s 2   p = - s + s 3

Omega Rank for B :  cycles: {{2, 9}, {6, 8}, {3, 5}},   net cycles: 3 .    order:   2

See Matrix
 

[0, y2, y1, 0, y2, y1, 0, y2, y1]

 

  p' = s 3 - s 5   p' = s - s 5   p' = s 2 - s 4   p = s - s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x5, x4, x3, x1, x2, -32 x5 - 16 x4 - 8 x3 - 4 x1 - 2 x2 ]
For A+2Δ :   [y1, -3 y1 - 3 y2 - y3 - 3 y7 - y6, -3 y1 - 3 y2 - 3 y7 - y4 - y5, y2, y3, y4, y7, y6, y5]
For A-2Δ :   [-3 y1 - 3 y4 - 3 y7 - y2 - y5, y1 - y3 + y4 - y6 + y7, y1, y2, y3, y4, y5, y6, y7]

Range of {ΩΔi}: [-μ1 - μ2, μ1, μ2, -μ1 - μ2, μ1, μ2, -μ1 - μ2, μ1, μ2]

 
rank of M is 9 , rank of N is 6

M              \ ;    N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $     $ [ [0, 3, 3, 6, 4, 6, 6, 5, 3] , [3, 0, 2, 4, 6, 4, 5, 6, 6] , [3, 2, 0, 6, 6, 6, 3, 4, 6] , [6, 4, 6, 0, 2, 0, 6, 6, 6] , [4, 6, 6, 2, 0, 2, 6, 6, 4] , [6, 4, 6, 0, 2, 0, 6, 6, 6] , [6, 5, 3, 6, 6, 6, 0, 1, 3] , [5, 6, 4, 6, 6, 6, 1, 0, 2] , [3, 6, 6, 6, 4, 6, 3, 2, 0] ] $

Check is ΩΔN zero? true, πΔ= [3, -2, -1, 3, -2, -1, 3, -2, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x4, x1, x2, x3, x5, x6, x7, x8, x9]

τ= 27 , r'= 2/3

Ranges

Action of R on ranges, [[1], [1], [1]]
Action of B on ranges, [[2], [3], [2]]
β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

ker N, [μ1, -μ1 - μ3, μ3, μ2, -μ1 - μ3, μ1 + μ3 - μ2, μ1, -μ1 - μ3, μ3]
Range of N
    [y1, y1 + y3 - y2 + y5 - y4, y1 + y5 - y6, y3, y2, y3, y5, y4, y6]

Partitions

Action of R on partitions, [[3], [3], [3]]
Action of B on partitions, [[2], [1], [1]]

α([{1, 5, 9}, {2, 4, 6}, {3, 7, 8}]) = 1/3
α([{1, 8, 9}, {2, 3, 7}, {4, 5, 6}]) = 1/6
α([{1, 2, 3}, {4, 5, 6}, {7, 8, 9}]) = 1/2

b1 = {1, 2, 3} ` , ` b2 = {1, 5, 9} ` , ` b3 = {1, 8, 9} ` , ` b4 = {2, 3, 7} ` , ` b5 = {2, 4, 6} ` , ` b6 = {3, 7, 8} ` , ` b7 = {4, 5, 6} ` , ` b8 = {7, 8, 9}

Action of R and B on the blocks of the partitions: = [8, 8, 8, 7, 1, 7, 1, 7] [2, 4, 5, 2, 3, 7, 6, 5]
with invariant measure [3, 2, 1, 1, 2, 2, 4, 3]

N by blocks, check: true . ` See partition graph.

` ` See level-3 partition graph.

`

Sandwich
Coloring {}
Rank3
R,B [4, 4, 4, 7, 7, 7, 1, 1, 1], [2, 9, 5, 8, 3, 8, 5, 6, 2]
π2 [0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0]
u2 [3, 3, 6, 4, 6, 6, 5, 3, 2, 4, 6, 4, 5, 6, 6, 6, 6, 6, 3, 4, 6, 2, 0, 6, 6, 6, 2, 6, 6, 4, 6, 6, 6, 1, 3, 2] (dim 1)
wpp [3, 3, 3, 3, 3, 3, 3, 3, 3]
π3 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u3 [2, 1, 1, 1, 2, 2, 0, 3, 1, 3, 0, 0, 0, 0, 0, 6, 5, 3, 0, 4, 3, 3, 6, 5, 3, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 4, 4, 0, 5, 6, 4, 3, 4, 4, 0, 2, 2, 2, 0, 3, 4, 6, 2, 3, 4, 4, 3, 4, 6, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 2, 2, 2, 0, 1, 1, 0, 1, 3, 2, 0]

 

 
2 . Coloring, {2}

R: [4, 9, 4, 7, 7, 7, 1, 1, 1]    B: [2, 4, 5, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )``]`

For τ=1/2, [-741, -228, -43, -645, -172, -43, -645, -172, -171] . FixedPtCheck, [741, 228, 43, 645, 172, 43, 645, 172, 171]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y3 - 3 y4 - 2 y6 + y1 + y5 - y2, 2 y3 + 2 y4 + y6 - y1 - y5, y1, y2, y3, y4, -y5 - y1, y5, y6]
  p = - s 3 + s 4 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 4 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

[y1, 0, 0, y4, 0, 0, y3, 0, y2]  

See Matrices
 

Omega Rank for B :  cycles: {{6, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 2 y2 - y4, y1, 2 y1 - y3, y2, y3, 0, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

 » SYNC'D 15/512 , 0.02929687500

 
3 . Coloring, {3}

R: [4, 4, 5, 7, 7, 7, 1, 1, 1]    B: [2, 9, 4, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-1470, -392, -114, -1425, -456, -95, -1482, -380, -98] . FixedPtCheck, [1470, 392, 114, 1425, 456, 95, 1482, 380, 98]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 2 vs 4 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

See Matrix
 

[y2, 0, 0, y2 - y1, y1, 0, y2, 0, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1 + y2 + y3 - y5, -y4 + y1 + y2 + y3, y1, y2, y3, 0, y4, y5]

 

  p' = s 4 - s 6   p = - s 4 + s 6

 » SYNC'D 15/4096 , 0.003662109375

 
4 . Coloring, {4}

R: [4, 4, 4, 8, 7, 7, 1, 1, 1]    B: [2, 9, 5, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )` 2 ` (` 5 - 4τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 4τ + τ 2 ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 - 4τ + τ 2 ` )``]`

For τ=1/2, [-585, -156, -19, -570, -76, -114, -285, -456, -39] . FixedPtCheck, [585, 156, 19, 570, 76, 114, 285, 456, 39]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 8 8 vs 8 2 vs 4 3 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

See Matrix
 

[y1 + y2, 0, 0, y1 + y2, 0, 0, y1, y2, 0]

 

  p = - s 2 + s 4   p = - s 2 + s 3

Omega Rank for B :  cycles: {{2, 9}, {6, 8}, {3, 5}},   net cycles: 2 .    order:   2

See Matrix
 

[0, 2 y1 - 4 y2, 2 y1 - 3 y2 - y3, 0, y1, y1 - 2 y2, y3, y2, 2 y2]

 

  p = s 2 - s 6   p' = s 2 - s 4   p' = s 3 - s 5   p' = - s 4 + s 6

 » SYNC'D 243/131072 , 0.001853942871

 
5 . Coloring, {5}

R: [4, 4, 4, 7, 3, 7, 1, 1, 1]    B: [2, 9, 5, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [255, 68, 57, 285, 76, 19, 247, 76, 17] . FixedPtCheck, [255, 68, 57, 285, 76, 19, 247, 76, 17]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y1 - 3 y2 - 3 y3 + 2 y5 + y4, 2 y2 + 2 y3 - 2 y5 - y4 - y6, -y5 - y4, y1, y2, y3, y4, y5, y6]
  p = s 3 - s 4 - 4s 5 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 4 2 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

[y2, 0, y1, y4, 0, 0, y3, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {6, 8}, {5, 7}},   net cycles: 3 .    order:   2

See Matrix
 

[0, y1, 0, 0, y1, y2, y2, y1, y2]

 

  p' = s - s 3   p = - s + s 5   p' = - s 3 + s 5   p = - s + s 3

 » SYNC'D 81/16384 , 0.004943847656

 
6 . Coloring, {6}

R: [4, 4, 4, 7, 7, 8, 1, 1, 1]    B: [2, 9, 5, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [1485, 396, 95, 1482, 380, 114, 1425, 456, 99] . FixedPtCheck, [1485, 396, 95, 1482, 380, 114, 1425, 456, 99]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 8 8 vs 8 2 vs 4 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

See Matrix
 

[y2 + y1, 0, 0, y2 + y1, 0, 0, y2, y1, 0]

 

  p' = s 2 - s 3   p = s 2 - s 4

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y2 + y1 + y5 - y4, y2, 0, y2 - y3 + y1 + y5, y3, y1, y5, y4]

 

  p' = s 4 - s 6   p = - s 4 + s 6

 » SYNC'D 1371/524288 , 0.002614974976

 
7 . Coloring, {7}

R: [4, 4, 4, 7, 7, 7, 5, 1, 1]    B: [2, 9, 5, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` - 5 + 3τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -27` (` - 1 + τ ` )` 3 ` (` - 5 + 3τ ` )``]`

For τ=1/2, [-210, -56, -114, -285, -456, -19, -570, -76, -14] . FixedPtCheck, [210, 56, 114, 285, 456, 19, 570, 76, 14]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2, y3 + 2 y5 + y4, -y3 - y5, y1, -y2 - y4, -y3 - 2 y5 - y4 - y1, y3, y5, y4]
  p = s 3 - 16s 5 + 8s 6 + 32s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s 3 - 4s 4 + 8s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 7 8 vs 8 8 vs 8 3 vs 4 3 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, 0, 0, y1, y2 - y1 + y3, 0, y3, 0, 0]

 

  p = s 3 - s 4

Omega Rank for B :  cycles: {{6, 8}, {2, 9}, {3, 5}},   net cycles: 2 .    order:   2

See Matrix
 

[2 y2 + y3 - y1, y2 + 2 y3, y2, 0, y3, 2 y3, 0, 2 y2, y1]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 675/262144 , 0.002574920654

 
8 . Coloring, {8}

R: [4, 4, 4, 7, 7, 7, 1, 6, 1]    B: [2, 9, 5, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [735, 196, 57, 741, 228, 171, 855, 228, 49] . FixedPtCheck, [735, 196, 57, 741, 228, 171, 855, 228, 49]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y4, y5, y3, y2, y1, -y5 - y1 - y4 - y2 - y6, -2 y4 - 2 y2 - 4 y1 - y5 - y6, 2 y4 - y3 + 2 y2 + 4 y1 + y5 + y6, y6]
  p = s 2 - 6s 4 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 4 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

[y3, 0, 0, y1, 0, y2, y4, 0, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y2 - y3, 2 y1 - y4, y2, 0, y1, 0, 0, y4, y3]

 

  p' = s 3 - s 5   p = - s 3 + s 5

 » SYNC'D 9/256 , 0.03515625000

 
9 . Coloring, {9}

R: [4, 4, 4, 7, 7, 7, 1, 1, 2]    B: [2, 9, 5, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-247, -76, -17, -255, -68, -17, -255, -68, -19] . FixedPtCheck, [247, 76, 17, 255, 68, 17, 255, 68, 19]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 2 vs 4 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: 0 .    order:   3

See Matrix
 

[-y1 + y2, y1, 0, y2, 0, 0, y2, 0, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}, {3, 5}},   net cycles: 3 .    order:   6

See Matrix
 

[y1, -y1 + y3 + y4 - y2, y3, 0, y4, y3, 0, y4, y2]

 

  p = - s + s 7   p = s - s 3 - s 4 + s 6   p = s + s 2 - s 4 - s 5

 » SYNC'D 3885/1048576 , 0.003705024719

 
10 . Coloring, {2, 3}

R: [4, 9, 5, 7, 7, 7, 1, 1, 1]    B: [2, 4, 4, 8, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [3822, 1176, 258, 3225, 1032, 215, 3354, 860, 882] . FixedPtCheck, [3822, 1176, 258, 3225, 1032, 215, 3354, 860, 882]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y3, y4, 0, y2, 0, 2 y4]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 4 y3, 3 y4, 3 y2, 3 y3, 3 y1, 0, 3 y5, 0]

 

  p = - s 4 + s 6

 » SYNC'D 15/256 , 0.05859375000

 
11 . Coloring, {2, 4}

R: [4, 9, 4, 8, 7, 7, 1, 1, 1]    B: [2, 4, 5, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )` 2 ` (` 5 - 4τ + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 4τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 ` (` 5 - 4τ + τ 2 ` )``]`

For τ=1/2, [-1521, -468, -43, -1290, -172, -258, -645, -1032, -351] . FixedPtCheck, [1521, 468, 43, 1290, 172, 258, 645, 1032, 351]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[2 y1, 0, 0, 2 y2, 0, 0, 3 y4, 2 y3, 2 y4]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{6, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[0, -y1 + 4 y3 + y5 - y4, y1, -y2 + y3 + 4 y5, y2, y3, y4, y5, 0]

 

  p' = s 4 - s 6   p = s 4 - s 6

 » SYNC'D 175/8192 , 0.02136230469

 
12 . Coloring, {2, 5}

R: [4, 9, 4, 7, 3, 7, 1, 1, 1]    B: [2, 4, 5, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-663, -204, -129, -645, -172, -43, -559, -172, -153] . FixedPtCheck, [663, 204, 129, 645, 172, 43, 559, 172, 153]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y6, y5, y3, y4, y2, y6 - 2 y3 + y4 - y2 - y1 + 2 y5, y1, -y3 - y1, -3 y5 + 2 y3 - 2 y6 - 2 y4 + y1]
  p = s 2 - 2s 4 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, y4, y3, 0, 0, y2, 0, y4]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{6, 8}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y4, 0, y3, y2, -y3 + 2 y1, y1, -y4 + 2 y2, 0]

 

  p = s 3 - s 5   p' = s 3 - s 5

 » SYNC'D 9/256 , 0.03515625000

 
13 . Coloring, {2, 6}

R: [4, 9, 4, 7, 7, 8, 1, 1, 1]    B: [2, 4, 5, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` 2 , 18` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 `]`

For τ=1/2, [3861, 1188, 215, 3354, 860, 258, 3225, 1032, 891] . FixedPtCheck, [3861, 1188, 215, 3354, 860, 258, 3225, 1032, 891]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y2, 0, 0, y3, y4, 2 y4]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[0, y5, y6, y4, y1, y2, y3, y7, 0]  

See Matrices
 

 » SYNC'D 495/8192 , 0.06042480469

 
14 . Coloring, {2, 7}

R: [4, 9, 4, 7, 7, 7, 5, 1, 1]    B: [2, 4, 5, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` - 5 + 3τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 54` (` - 5 + 3τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 27` (` - 5 + 3τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-546, -168, -258, -645, -1032, -43, -1290, -172, -126] . FixedPtCheck, [546, 168, 258, 645, 1032, 43, 1290, 172, 126]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y3 + y1 - y2 + y4, 0, 0, y3, y1, 0, y2, 0, y4]

 

  p = s 4 - s 5

Omega Rank for B :  cycles: {{6, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - y2 + 3 y4 - y3, 3 y1 + 2 y4 - y5, y1, y2, y4, y3, 0, y5, 0]

 

  p' = - s 4 + s 6   p = s 4 - s 6

 » SYNC'D 1571/65536 , 0.02397155762

 
15 . Coloring, {2, 8}

R: [4, 9, 4, 7, 7, 7, 1, 6, 1]    B: [2, 4, 5, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-637, -196, -43, -559, -172, -129, -645, -172, -147] . FixedPtCheck, [637, 196, 43, 559, 172, 129, 645, 172, 147]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y1, 2 y2 - 2 y3 + y4 + y6, -y4 - y5, -y1 - 3 y2 + y3 - y4 - 2 y6, y2, y3, y4, y5, y6]
  p = s 3 + s 4 - 4s 5 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 2 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y4, 0, y3, y2, 0, y3]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {3, 5}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y2, y1, y1, y2, 0, 0, y2, 0]

 

  p' = s - s 3   p' = s 2 - s 4   p = s - s 5   p' = - s 3 + s 5

 » SYNC'D 675/16384 , 0.04119873047

 
16 . Coloring, {2, 9}

R: [4, 9, 4, 7, 7, 7, 1, 1, 2]    B: [2, 4, 5, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-399, -228, -25, -375, -100, -25, -375, -100, -171] . FixedPtCheck, [399, 228, 25, 375, 100, 25, 375, 100, 171]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 2 .    order:   6

See Matrix
 

[y2, y3, 0, y1, 0, 0, -y2 + 5 y3 - y1 + 5 y4, 0, y4]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{6, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y5 - 4 y4 - y3 + 3 y2, y1, 2 y1 - 3 y4 + 2 y2, y5, y4, y3, 0, y2, 0]

 

  p' = s 4 - s 6   p = s 4 - s 6

 » SYNC'D 1145/131072 , 0.008735656738

 
17 . Coloring, {3, 4}

R: [4, 4, 5, 8, 7, 7, 1, 1, 1]    B: [2, 9, 4, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [495, 132, 19, 475, 76, 95, 247, 380, 33] . FixedPtCheck, [495, 132, 19, 475, 76, 95, 247, 380, 33]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 4 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 3 vs 5 4 vs 8

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

See Matrix
 

[y2 + y3, 0, 0, -y1 + y2 + y3, y1, 0, y2, y3, 0]

 

  p = s 3 - s 5   p' = s 3 - s 4

Omega Rank for B :  cycles: {{2, 9}, {3, 4, 5, 7}, {6, 8}},   net cycles: 3 .    order:   4

See Matrix
 

[0, 2 y3, 2 y3 + y2 - y4, -y1 + y3 + 2 y2, y1, y3, y4, y2, 2 y2]

 

  p' = s - s 5   p = - s + s 5   p' = - s 2 + s 6   p' = - s 3 + s 7

 » SYNC'D 155043/33554432 , 0.004620641470

 
18 . Coloring, {3, 5}

R: [4, 4, 5, 7, 3, 7, 1, 1, 1]    B: [2, 9, 4, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-270, -72, -114, -285, -152, -19, -266, -76, -18] . FixedPtCheck, [270, 72, 114, 285, 152, 19, 266, 76, 18]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 3 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 2 .    order:   6

See Matrix
 

[5 y1 - y2 + 5 y3 - y4, 0, y1, y2, y3, 0, y4, 0, 0]

 

  p = s + s 2 - s 4 - s 5

Omega Rank for B :  cycles: {{2, 9}, {6, 8}, {5, 7}},   net cycles: 2 .    order:   2

See Matrix
 

[0, 8 y2 + 8 y1 - 10 y3, 0, y2, 5 y2 + 5 y1 - 6 y3, y1, y3, 6 y2 + 6 y1 - 7 y3, -2 y2 - 2 y1 + 4 y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 645/524288 , 0.001230239868

 
19 . Coloring, {3, 6}

R: [4, 4, 5, 7, 7, 8, 1, 1, 1]    B: [2, 9, 4, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-255, -68, -19, -247, -76, -19, -247, -76, -17] . FixedPtCheck, [255, 68, 19, 247, 76, 19, 247, 76, 17]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 4 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 2 vs 5 4 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1 + y2, 0, 0, y1, y2, 0, y1, y2, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 9}, {3, 4, 5, 6, 7, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[0, y4, y3, y2, y1, y3, y2, y1, -y4 + y2 + y1 + y3]

 

  p' = s 3 + s 4 - s 6 - s 7   p' = s 2 - s 4 - s 5 + s 7   p' = s - s 7   p = s - s 7

 » SYNC'D 209385/33554432 , 0.006240159273

 
20 . Coloring, {3, 7}

R: [4, 4, 5, 7, 7, 7, 5, 1, 1]    B: [2, 9, 4, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-255, -68, -171, -285, -684, -19, -741, -76, -17] . FixedPtCheck, [255, 68, 171, 285, 684, 19, 741, 76, 17]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y4 - y1 + y2 - 3 y3, y4 - 2 y2 + 2 y3 - y6, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 3 - s 4 + 4s 5 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 3 vs 4 4 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1, 0, 0, y1 - y2 + y3, y2, 0, y3, 0, 0]

 

  p = s 3 - s 4

Omega Rank for B :  cycles: {{2, 9}, {6, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[3 y3, 2 y4, 2 y3, 2 y2, 0, 2 y1, 0, 6 y4 - 2 y3 - 4 y2 - 4 y1, 4 y4 - 2 y2 - 2 y1 - 3 y3]

 

  p' = s 3 - s 5   p = s 3 - s 7   p' = s 4 - s 6

 » SYNC'D 69/4096 , 0.01684570312

 
21 . Coloring, {3, 8}

R: [4, 4, 5, 7, 7, 7, 1, 6, 1]    B: [2, 9, 4, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (` - 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` 2 , -18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -27` (` - 1 + τ ` )` 2 ` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [-3810, -1016, -342, -3705, -1368, -855, -4446, -1140, -254] . FixedPtCheck, [3810, 1016, 342, 3705, 1368, 855, 4446, 1140, 254]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y3, y2, 2 y2, y4, 0, 0]

 

  p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   6

[y2, y1, y5, y4, y3, 0, 0, y6, y7]  

See Matrices
 

 » SYNC'D 15005/262144 , 0.05723953247

 
22 . Coloring, {3, 9}

R: [4, 4, 5, 7, 7, 7, 1, 1, 2]    B: [2, 9, 4, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-1274, -392, -102, -1275, -408, -85, -1326, -340, -98] . FixedPtCheck, [1274, 392, 102, 1275, 408, 85, 1326, 340, 98]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 2 vs 5 6 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y1, 0, y2, y1, 0, y2 + y1, 0, 0]

 

  p = s 2 - s 3   p' = - s 2 + s 3   p' = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y6, y5, y3, y4, y2, y1, 0, -y3 + y4 + y2 + y1, -y6 - y5 + y4 + y2 + y1]

 

  p' = - s 4 + s 7   p = s 4 - s 7

 » SYNC'D 675/262144 , 0.002574920654

 
23 . Coloring, {4, 5}

R: [4, 4, 4, 8, 3, 7, 1, 1, 1]    B: [2, 9, 5, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [555, 148, 57, 570, 76, 114, 247, 456, 37] . FixedPtCheck, [555, 148, 57, 570, 76, 114, 247, 456, 37]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 2 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 2 y3, y2, 0, 0, y3, y4, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 9}, {5, 7}, {6, 8}},   net cycles: 3 .    order:   2

See Matrix
 

[0, 2 y2 - 4 y1, 0, 0, y2, y2 - 2 y1, 2 y2 - 3 y1, y1, 2 y1]

 

  p = s - s 5   p' = s - s 3   p' = s 2 - s 4   p' = - s 3 + s 5

 » SYNC'D 171/32768 , 0.005218505859

 
24 . Coloring, {4, 6}

R: [4, 4, 4, 8, 7, 8, 1, 1, 1]    B: [2, 9, 5, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-765, -204, -19, -741, -76, -171, -285, -684, -51] . FixedPtCheck, [765, 204, 19, 741, 76, 171, 285, 684, 51]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y3, y4, -3 y4 - 2 y3 - 2 y2 - 3 y5 + y6, y2, y1, -y3 - y4 - y2 - y1 - y5, 3 y4 + 2 y3 + 2 y2 + 3 y5 - 2 y6, y6, y5]
  p = s 2 + 2s 4 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 6 vs 7 7 vs 7 2 vs 4 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

See Matrix
 

[y2 + y1, 0, 0, y2 + y1, 0, 0, y2, y1, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4

Omega Rank for B :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1 + y2 - y4, y1 + y2 - y3, 0, y1, y2, y3, 0, y4]

 

  p' = s 3 - s 5   p = - s 3 + s 5

 » SYNC'D 69/8192 , 0.008422851562

 
25 . Coloring, {4, 7}

R: [4, 4, 4, 8, 7, 7, 5, 1, 1]    B: [2, 9, 5, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [255, 68, 57, 285, 228, 57, 285, 228, 17] . FixedPtCheck, [255, 68, 57, 285, 228, 57, 285, 228, 17]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )`` (` - 1 + τ ` )` 4

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y3 + y4 - 2 y2, y5 + y3 - y1 + y2 - y4, -y5 - y3, y1, -y4 - y5, y2, y5, y3, y4]
  p = s - 4s 3 - 4s 4 + 4s 5 + 8s 6

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s - 4s 3 - 4s 4 + 4s 5 + 8s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 7 9 vs 9 9 vs 9 3 vs 5 4 vs 8

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y1, 0, 0, -y1 + 4 y2 - y3, y2, 0, y2, y3, 0]

 

  p = - s + s 4   p' = - s + s 4

Omega Rank for B :  cycles: {{3, 5}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y1 + 3 y3 - y4, 3 y1 - y2 + y3, y1, 0, y3, y1, y2, y3, y4]

 

  p' = s 3 - s 5   p = - s 3 + s 5   p = - s 3 + s 7   p' = - s 5 + s 7

 » SYNC'D 4725/2097152 , 0.002253055573

 
26 . Coloring, {4, 8}

R: [4, 4, 4, 8, 7, 7, 1, 6, 1]    B: [2, 9, 5, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [1485, 396, 95, 1482, 380, 1026, 1425, 1368, 99] . FixedPtCheck, [1485, 396, 95, 1482, 380, 1026, 1425, 1368, 99]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 5 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: 1 .    order:   5

[y5, 0, 0, y4, 0, y3, y2, y1, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{3, 5}, {2, 9}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, y1 + 2 y3, y1, 0, y2 + y4, 0, 3 y3, y3, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 725/65536 , 0.01106262207

 
27 . Coloring, {4, 9}

R: [4, 4, 4, 8, 7, 7, 1, 1, 2]    B: [2, 9, 5, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [507, 156, 17, 510, 68, 102, 255, 408, 39] . FixedPtCheck, [507, 156, 17, 510, 68, 102, 255, 408, 39]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )`` (` - 1 + τ ` )` 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 2 vs 5 5 vs 8

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[2 y1 + y2, y1, 0, 3 y1 + y2, 0, 0, 3 y1, y2, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y2, -y2 + 2 y1 + 2 y4 - y5, 2 y1 + y4 - y3, 0, y1 + 2 y4, y1, y3, y4, y5]

 

  p = - s 2 + s 8   p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

 » SYNC'D 53265/8388608 , 0.006349682808

 
28 . Coloring, {5, 6}

R: [4, 4, 4, 7, 3, 8, 1, 1, 1]    B: [2, 9, 5, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [-1335, -356, -285, -1482, -380, -114, -1235, -456, -89] . FixedPtCheck, [1335, 356, 285, 1482, 380, 114, 1235, 456, 89]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 2 y4, y2, 0, 0, y3, y4, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y3, 0, 0, y2, y3 - y2 + y4, y3 - y1 + y4, y1, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

 » SYNC'D 99/4096 , 0.02416992188

 
29 . Coloring, {5, 7}

R: [4, 4, 4, 7, 3, 7, 5, 1, 1]    B: [2, 9, 5, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -54` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-690, -184, -1026, -1425, -1368, -95, -1482, -380, -46] . FixedPtCheck, [690, 184, 1026, 1425, 1368, 95, 1482, 380, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[-y2 + y3 + y1 - y4, 0, y2, y3, y1, 0, y4, 0, 0]

 

  p = s 2 - s 3 + s 4 - s 5

Omega Rank for B :  cycles: {{6, 8}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[y4 + y3 - y1 - y2, y4 - y5 + y3, 0, 0, y1, y4, y5, y3, y2]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

 » SYNC'D 4815/524288 , 0.009183883667

 
30 . Coloring, {5, 8}

R: [4, 4, 4, 7, 3, 7, 1, 6, 1]    B: [2, 9, 5, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [645, 172, 171, 741, 228, 171, 741, 228, 43] . FixedPtCheck, [645, 172, 171, 741, 228, 171, 741, 228, 43]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y2 + y3 + 2 y5 + y4 - y1, 2 y2 - 2 y3 - y6 - 2 y5 - y4, -y5 - y4, y1, y2, y3, y4, y5, y6]
  p = s 3 - 3s 4 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, y3, y2, 0, y3, y4, 0, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y2 - y4, 2 y1 - y3, 0, 0, y1, 0, y2, y3, y4]

 

  p' = s 3 - s 5   p = - s 3 + s 5

 » SYNC'D 15/512 , 0.02929687500

 
31 . Coloring, {5, 9}

R: [4, 4, 4, 7, 3, 7, 1, 1, 2]    B: [2, 9, 5, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [13, 4, 3, 15, 4, 1, 13, 4, 1] . FixedPtCheck, [13, 4, 3, 15, 4, 1, 13, 4, 1]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 2 y2, y3, 0, 0, y4, 0, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{6, 8}, {5, 7}, {1, 2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[-y1 + y3 + y4 - y2, y1, 0, 0, y3, y4, y4, y3, y2]

 

  p = s - s 3 - s 4 + s 6   p = - s + s 7   p = - s - s 2 + s 4 + s 5

 » SYNC'D 30495/2097152 , 0.01454114914

 
32 . Coloring, {6, 7}

R: [4, 4, 4, 7, 7, 8, 5, 1, 1]    B: [2, 9, 5, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [555, 148, 285, 741, 1140, 57, 1425, 228, 37] . FixedPtCheck, [555, 148, 285, 741, 1140, 57, 1425, 228, 37]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 4 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 5 6 vs 8

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y5, 0, 0, y3, y4, 0, y1, y2, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y3, y2, 0, y1, -y4 + 2 y2 + 3 y1 - y5, -y3 + 3 y2 + 2 y1 - y6, y6, y5]

 

  p' = s 5 - s 7   p = s 5 - s 7

 » SYNC'D 128899/8388608 , 0.01536595821

 
33 . Coloring, {6, 8}

R: [4, 4, 4, 7, 7, 8, 1, 6, 1]    B: [2, 9, 5, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` - 5 - 3τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [795, 212, 57, 798, 228, 342, 855, 456, 53] . FixedPtCheck, [795, 212, 57, 798, 228, 342, 855, 456, 53]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[-y1 + 5 y2 - y3 + 5 y4, 0, 0, y1, 0, y2, y3, y4, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 0 .    order:   4

See Matrix
 

[-16 y2 + 33 y1 - 48 y4 - 5 y3, 5 y2, -7 y2 + 16 y1 - 26 y4, 0, 5 y1, 0, 5 y4, 15 y4, 5 y3]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 957/65536 , 0.01460266113

 
34 . Coloring, {6, 9}

R: [4, 4, 4, 7, 7, 8, 1, 1, 2]    B: [2, 9, 5, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-1287, -396, -85, -1326, -340, -102, -1275, -408, -99] . FixedPtCheck, [1287, 396, 85, 1326, 340, 102, 1275, 408, 99]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 4 ` (` 1 + τ ` )`` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 2 vs 5 6 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y1 + y2, 0, 0, y1, y2, 0]

 

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + y2 + y3 - y6, y1, y2 + y3 - y4 - y5, 0, y2, y3, y4, y5, y6]

 

  p' = - s 4 + s 7   p = - s 4 + s 7

 » SYNC'D 213555/33554432 , 0.006364434958

 
35 . Coloring, {7, 8}

R: [4, 4, 4, 7, 7, 7, 5, 6, 1]    B: [2, 9, 5, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [510, 136, 342, 741, 1368, 171, 1710, 228, 34] . FixedPtCheck, [510, 136, 342, 741, 1368, 171, 1710, 228, 34]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, y2, y3, 2 y1, y4, 0, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y2 + 3 y3 - y4, 3 y2 + 2 y3 - y1, y2, 0, y3, 0, 0, y1, y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

 » SYNC'D 705/16384 , 0.04302978516

 
36 . Coloring, {7, 9}

R: [4, 4, 4, 7, 7, 7, 5, 1, 2]    B: [2, 9, 5, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 5 + 3τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -54` (` - 5 + 3τ ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 27` (` - 5 + 3τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [182, 56, 102, 255, 408, 17, 510, 68, 14] . FixedPtCheck, [182, 56, 102, 255, 408, 17, 510, 68, 14]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 3 vs 5 4 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y1, y1, 0, 3 y1 - y2 + y3, y2, 0, y3, 0, 0]

 

  p = s 3 - s 5   p' = s 3 - s 4

Omega Rank for B :  cycles: {{3, 5}, {6, 8}, {1, 2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[-y1 + 3 y2 + 3 y3 - y4, y1, y2, 0, y3, 2 y3, 0, 2 y2, y4]

 

  p' = s + s 2 - s 4 - s 5   p = - s - s 2 + s 4 + s 5   p = - s + s 7

 » SYNC'D 6591/524288 , 0.01257133484

 
37 . Coloring, {8, 9}

R: [4, 4, 4, 7, 7, 7, 1, 6, 2]    B: [2, 9, 5, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-637, -196, -51, -663, -204, -153, -765, -204, -49] . FixedPtCheck, [637, 196, 51, 663, 204, 153, 765, 204, 49]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y3, 0, 2 y2, y4, 0, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 2 y2 + 2 y3 - y4 - y5, y1, y2, 0, y3, 0, 0, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 2267/32768 , 0.06918334961

 
38 . Coloring, {2, 3, 4}

R: [4, 9, 5, 8, 7, 7, 1, 1, 1]    B: [2, 4, 4, 7, 3, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-1287, -396, -43, -1075, -172, -215, -559, -860, -297] . FixedPtCheck, [1287, 396, 43, 1075, 172, 215, 559, 860, 297]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y4, y3, 0, y2, y5, 2 y3]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y2, -y2 + 4 y3 - y4 + y5, y1, -y1 + y3 + 4 y5, y3, y4, y5, 0]

 

  p' = s 2 - s 6   p = s 2 - s 6

 » SYNC'D 16875/524288 , 0.03218650818

 
39 . Coloring, {2, 3, 5}

R: [4, 9, 5, 7, 3, 7, 1, 1, 1]    B: [2, 4, 4, 8, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [702, 216, 258, 645, 344, 43, 602, 172, 162] . FixedPtCheck, [702, 216, 258, 645, 344, 43, 602, 172, 162]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y3 + 5 y2 - y4 - y5, 0, y1, y3, y2, 0, y4, 0, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{6, 8}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[0, -14 y4 - 14 y3 + 39 y2 - y1, 0, y4, -5 y4 - 5 y3 + 14 y2, y3, y2, y1, 0]

 

  p' = s 3 - s 5   p = s 3 - s 5

 » SYNC'D 51/4096 , 0.01245117188

 
40 . Coloring, {2, 3, 6}

R: [4, 9, 5, 7, 7, 8, 1, 1, 1]    B: [2, 4, 4, 8, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [663, 204, 43, 559, 172, 43, 559, 172, 153] . FixedPtCheck, [663, 204, 43, 559, 172, 43, 559, 172, 153]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y2, 0, 0, y1, y4, 0, y3, y4, 2 y4]

 

  p = s 2 - s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 8}},   net cycles: 0 .    order:   6

[0, y1, y2, y3, y4, y5, y6, y7, 0]  

See Matrices
 

 » SYNC'D 82215/2097152 , 0.03920316696

 
41 . Coloring, {2, 3, 7}

R: [4, 9, 5, 7, 7, 7, 5, 1, 1]    B: [2, 4, 4, 8, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [663, 204, 387, 645, 1548, 43, 1677, 172, 153] . FixedPtCheck, [663, 204, 387, 645, 1548, 43, 1677, 172, 153]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2 - 3 y3 - y4 - 2 y6 - y1, -2 y2 + 2 y3 + y4 + y6, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 2 + 6s 4 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1, 0, 0, y1 - y2 + y4 - y3, y2, 0, y4, 0, y3]

 

  p = - s 4 + s 5

Omega Rank for B :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[3 y4, 2 y5, 2 y4, 2 y3, 0, 2 y2, 0, 2 y1, 0]

 

  p = s 4 - s 6

 » SYNC'D 551/8192 , 0.06726074219

 
42 . Coloring, {2, 3, 8}

R: [4, 9, 5, 7, 7, 7, 1, 6, 1]    B: [2, 4, 4, 8, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 3 + τ 2 ` )` , 54` (` - 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 27` (` - 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-3302, -1016, -258, -2795, -1032, -645, -3354, -860, -762] . FixedPtCheck, [3302, 1016, 258, 2795, 1032, 645, 3354, 860, 762]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y4, 0, 0, y3, y2, 2 y2, y1, 0, 2 y2]

 

  p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y3, y1, y2, y6, y4, 0, 0, y5, 0]  

See Matrices
 

 » SYNC'D 3645/131072 , 0.02780914307

 
43 . Coloring, {2, 3, 9}

R: [4, 9, 5, 7, 7, 7, 1, 1, 2]    B: [2, 4, 4, 8, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-2058, -1176, -150, -1875, -600, -125, -1950, -500, -882] . FixedPtCheck, [2058, 1176, 150, 1875, 600, 125, 1950, 500, 882]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y1, 2 y1 + y3 + y6 - 2 y5, -y4 - y6, -3 y1 - 2 y3 - y6 + y5 - y2, y3, y2, y4, y6, y5]
  p = s 2 + 2s 3 - 4s 5 - 8s 6 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y3 - y4 + 5 y5, y1, 0, y2, y3, 0, y4, 0, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, -7 y1 + 3 y2, -11 y1 + 4 y2 - y3 + y4, 3 y1, y3, 0, y4, 0]

 

  p = - s 4 + s 7   p = - s 4 + s 5   p = - s 4 + s 6

 » SYNC'D 6645/262144 , 0.02534866333

 
44 . Coloring, {2, 4, 5}

R: [4, 9, 4, 8, 3, 7, 1, 1, 1]    B: [2, 4, 5, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1443, 444, 129, 1290, 172, 258, 559, 1032, 333] . FixedPtCheck, [1443, 444, 129, 1290, 172, 258, 559, 1032, 333]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, 0, 2 y3, y2, 0, 0, y3, y4, 2 y3]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 4 y1 + 4 y4 - 15 y3 - y2, 0, y1, y4, y1 + y4 - 4 y3, y2, y3, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

 » SYNC'D 123/4096 , 0.03002929688

 
45 . Coloring, {2, 4, 6}

R: [4, 9, 4, 8, 7, 8, 1, 1, 1]    B: [2, 4, 5, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 `]`

For τ=1/2, [1989, 612, 43, 1677, 172, 387, 645, 1548, 459] . FixedPtCheck, [1989, 612, 43, 1677, 172, 387, 645, 1548, 459]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y1 - y2 - y6 - y3 - y4, y1, -y1 + y6 + 2 y3 + 2 y4 + y5, y2, y3, y4, y1 - y6 - 2 y3 - 2 y4 - 2 y5, y5, y6]
  p = s 3 + s 4 + 4s 5 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y4, 0, 0, y3, y2, y3]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 2 y5, y1, y2, y3, y5, y4, 0, 0]

 

  p = - s 4 + s 6

 » SYNC'D 175/4096 , 0.04272460938

 
46 . Coloring, {2, 4, 7}

R: [4, 9, 4, 8, 7, 7, 5, 1, 1]    B: [2, 4, 5, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-221, -68, -43, -215, -172, -43, -215, -172, -51] . FixedPtCheck, [221, 68, 43, 215, 172, 43, 215, 172, 51]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 8

Omega Rank for R :  cycles: {{1, 4, 8}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-y3 + 4 y2 - y1 - y4, 0, 0, y3, y2, 0, y2, y1, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}, {3, 5}},   net cycles: 3 .    order:   4

See Matrix
 

[y1 - y3 + 3 y4, 3 y1 + y4 - y2, y1, y3, y4, y1, y2, y4, 0]

 

  p' = s 3 - s 7   p = - s + s 5   p' = - s + s 5   p' = s 2 - s 6

 » SYNC'D 24105/16777216 , 0.001436769962

 
47 . Coloring, {2, 4, 8}

R: [4, 9, 4, 8, 7, 7, 1, 6, 1]    B: [2, 4, 5, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [3861, 1188, 215, 3354, 860, 2322, 3225, 3096, 891] . FixedPtCheck, [3861, 1188, 215, 3354, 860, 2322, 3225, 3096, 891]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: 0 .    order:   5

[y2, 0, 0, y1, 0, y6, y5, y4, y3]  

See Matrices
 

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[y1, y2, y3, y4, y5, 0, y6, y7, 0]  

See Matrices
 

 » SYNC'D 18335/1048576 , 0.01748561859

 
48 . Coloring, {2, 4, 9}

R: [4, 9, 4, 8, 7, 7, 1, 1, 2]    B: [2, 4, 5, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 18` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 - 4τ + τ 2 ` )`` (` 1 + τ ` )` 3 `]`

For τ=1/2, [819, 468, 25, 750, 100, 150, 375, 600, 351] . FixedPtCheck, [819, 468, 25, 750, 100, 150, 375, 600, 351]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-2 y1 - 3 y2 - 3 y6 - y5 - y3, y1 + 2 y2 + 2 y6 + y5, -y5 - y4, y3, y1, y2, y4, y5, y6]
  p = s 2 - 2s 3 + 4s 5 + 8s 6 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 9 vs 9 9 vs 9 5 vs 6 6 vs 8

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y1, 0, -y2 + 5 y1 - y5 - y4 + 5 y3, 0, 0, y5, y4, y3]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{6, 8}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, -y1 + 3 y4 - y6 + 2 y5, y1, -y2 - y3 + 2 y4 + 3 y5, y3, y4, y6, y5, 0]

 

  p' = s 5 - s 7   p = s 5 - s 7

 » SYNC'D 96495/16777216 , 0.005751550198

 
49 . Coloring, {2, 5, 6}

R: [4, 9, 4, 7, 3, 8, 1, 1, 1]    B: [2, 4, 5, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )``]`

For τ=1/2, [-3471, -1068, -645, -3354, -860, -258, -2795, -1032, -801] . FixedPtCheck, [3471, 1068, 645, 3354, 860, 258, 2795, 1032, 801]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, 0, 2 y3, y2, 0, 0, y4, y3, 2 y3]

 

  p' = - s 2 + s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[0, y3, 0, y4, y2, y1, y5, y6, 0]  

See Matrices
 

 » SYNC'D 141/2048 , 0.06884765625

 
50 . Coloring, {2, 5, 7}

R: [4, 9, 4, 7, 3, 7, 5, 1, 1]    B: [2, 4, 5, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` 3 + τ 2 ` )` , 54` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 27` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [1794, 552, 2322, 3225, 3096, 215, 3354, 860, 414] . FixedPtCheck, [1794, 552, 2322, 3225, 3096, 215, 3354, 860, 414]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, 0, y3, y2 + y3 - y1 + y4 - y5, y1, 0, y4, 0, y5]

 

  p = - s 3 + s 4 - s 5 + s 6

Omega Rank for B :  cycles: {{6, 8}},   net cycles: 0 .    order:   6

[y1, y2, 0, y3, y4, y5, y6, y7, 0]  

See Matrices
 

 » SYNC'D 35301/1048576 , 0.03366565704

 
51 . Coloring, {2, 5, 8}

R: [4, 9, 4, 7, 3, 7, 1, 6, 1]    B: [2, 4, 5, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-3` (` 3 + τ 2 ` )` , 6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )`` (` 1 + τ ` )` , -3` (` 3 + τ 2 ` )` , 6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )`` (` 1 + τ ` )` , -3` (` 3 + τ 2 ` )` , 6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-13, -4, -3, -13, -4, -3, -13, -4, -3] . FixedPtCheck, [13, 4, 3, 13, 4, 3, 13, 4, 3]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y1 - y2, y1, y2, -y1 - y2, y1, y2, -y1 - y2, y1, y2]
  p = s 2 - 32s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s 2 - 16s 6   p' = s 3 - 8s 6   p' = s 4 - 4s 6   p' = s 5 - 2s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 7 2 vs 7 2 vs 7 2 vs 6 2 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, 0, y2, y1, 0, y2, y1, 0, y2]

 

  p = s 2 - s 5   p' = s 2 - s 4   p' = s 3 - s 4   p' = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y2, 0, y1, y2, 0, y1, y2, 0]

 

  p = - s + s 3   p' = - s + s 3   p = - s + s 5   p' = - s + s 5


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[0, x5, x4, x3, x2, x1, -32 x5 - 16 x4 - 8 x3 - 4 x2 - 2 x1 ]
For A+2Δ :   [-3 y1 - y4 - 3 y2 - y5 - 3 y6, 9 y1 + 9 y2 + 9 y6 - y3 - y7, y1, y4, y3, y2, y5, y7, y6]
For A-2Δ :   [y5, y4, y3, y2, y1, 9 y4 - y3 + 9 y1 + 9 y7 - y6, -y5 - 3 y4 - y2 - 3 y1 - 3 y7, y7, y6]

Range of {ΩΔi}: [-μ2 - μ1, μ2, μ1, -μ2 - μ1, μ2, μ1, -μ2 - μ1, μ2, μ1]

 
rank of M is 9 , rank of N is 6

M              \ ;    N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $     $ [ [0, 16, 10, 20, 16, 20, 20, 8, 10] , [16, 0, 17, 12, 20, 12, 12, 20, 11] , [10, 17, 0, 20, 15, 20, 10, 8, 20] , [20, 12, 20, 0, 14, 0, 20, 14, 20] , [16, 20, 15, 14, 0, 14, 10, 20, 11] , [20, 12, 20, 0, 14, 0, 20, 14, 20] , [20, 12, 10, 20, 10, 20, 0, 18, 10] , [8, 20, 8, 14, 20, 14, 18, 0, 18] , [10, 11, 20, 20, 11, 20, 10, 18, 0] ] $

Check is ΩΔN zero? true, πΔ= [1, -2, 1, 1, -2, 1, 1, -2, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8, x9]

τ= 27 , r'= 2/3

Ranges

Action of R on ranges, [[1], [3], [1]]
Action of B on ranges, [[2], [1], [2]]
β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

ker N, [μ1, μ2, -μ1 - μ2, -μ2 - μ3, μ2, μ3, μ1, μ2, -μ1 - μ2]
Range of N
    [y6, y3, y4, y5, y1, y5, y2, y6 - y3 + y5 - y1 + y2, y6 - y4 + y2]

Partitions

Action of R on partitions, [[2], [3], [3], [2], [2], [2], [2], [2]]
Action of B on partitions, [[4], [6], [7], [8], [7], [1], [5], [6]]

α([{1, 5, 9}, {2, 4, 6}, {3, 7, 8}]) = 1/15
α([{1, 3, 8}, {2, 7, 9}, {4, 5, 6}]) = 1/4
α([{1, 3, 8}, {2, 4, 6}, {5, 7, 9}]) = 1/4
α([{1, 2, 9}, {3, 7, 8}, {4, 5, 6}]) = 1/30
α([{1, 8, 9}, {2, 4, 6}, {3, 5, 7}]) = 1/12
α([{1, 5, 9}, {2, 3, 7}, {4, 6, 8}]) = 2/15
α([{1, 2, 9}, {3, 5, 7}, {4, 6, 8}]) = 1/6
α([{1, 8, 9}, {2, 3, 7}, {4, 5, 6}]) = 1/60

b1 = {1, 2, 9} ` , ` b2 = {1, 3, 8} ` , ` b3 = {1, 5, 9} ` , ` b4 = {1, 8, 9} ` , ` b5 = {2, 3, 7} ` , ` b6 = {2, 4, 6} ` , ` b7 = {2, 7, 9} ` , ` b8 = {3, 5, 7} ` , ` b9 = {3, 7, 8} ` , ` b10 = {4, 5, 6} ` , ` b11 = {4, 6, 8} ` , ` b12 = {5, 7, 9}

Action of R and B on the blocks of the partitions: = [7, C, 7, 7, A, 2, 6, A, A, 2, 2, 6] [4, B, 9, B, 3, 1, 3, 8, A, 5, 6, 8]
with invariant measure [ 3 5 5 5] [2, 5, 2, 1, -, 4, -, -, 1, 3, 3, -] [ 2 2 2 2]

N by blocks, check: true . ` See partition graph.

` ` See level-3 partition graph.

`

Sandwich
Coloring {2, 5, 8}
Rank3
R,B [4, 9, 4, 7, 3, 7, 1, 6, 1], [2, 4, 5, 8, 7, 8, 5, 1, 2]
π2 [0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0]
u2 [16, 10, 20, 16, 20, 20, 8, 10, 17, 12, 20, 12, 12, 20, 11, 20, 15, 20, 10, 8, 20, 14, 0, 20, 14, 20, 14, 10, 20, 11, 20, 14, 20, 18, 10, 18] (dim 1)
wpp [3, 3, 3, 3, 3, 3, 3, 3, 3]
π3 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u3 [3, 8, 12, 8, 8, 4, 5, 10, 1, 10, 0, 6, 0, 10, 0, 20, 2, 10, 10, 6, 4, 5, 20, 2, 10, 6, 0, 0, 9, 12, 9, 5, 5, 8, 6, 0, 4, 6, 3, 6, 2, 20, 2, 4, 6, 3, 10, 3, 9, 9, 0, 10, 2, 20, 9, 5, 3, 6, 10, 2, 20, 0, 0, 6, 0, 4, 8, 5, 0, 0, 0, 12, 10, 12, 4, 8, 5, 8, 1, 9, 12, 10, 12, 6]

 

 
52 . Coloring, {2, 5, 9}

R: [4, 9, 4, 7, 3, 7, 1, 1, 2]    B: [2, 4, 5, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [357, 204, 75, 375, 100, 25, 325, 100, 153] . FixedPtCheck, [357, 204, 75, 375, 100, 25, 325, 100, 153]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y3 - y4 + 5 y5, y1, y2, y3, 0, 0, y4, 0, y5]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{6, 8}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y2 - 4 y3 - y4 + 3 y5, y1, 0, y2, y3, y4, 2 y1 - 3 y3 + 2 y5, y5, 0]

 

  p = s 4 - s 6   p' = - s 4 + s 6

 » SYNC'D 3645/262144 , 0.01390457153

 
53 . Coloring, {2, 6, 7}

R: [4, 9, 4, 7, 7, 8, 5, 1, 1]    B: [2, 4, 5, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-481, -148, -215, -559, -860, -43, -1075, -172, -111] . FixedPtCheck, [481, 148, 215, 559, 860, 43, 1075, 172, 111]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 6 vs 8

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y5, 0, 0, y3, y4, 0, y1, y2, 2 y2]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 6, 7, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[2 y1 - y2 + 3 y3 - y6, 3 y1 + 2 y3 - y4 - y5, y1, y2, y3, y6, y4, y5, 0]

 

  p = - s + s 7   p' = - s + s 7

 » SYNC'D 1176035/33554432 , 0.03504857421

 
54 . Coloring, {2, 6, 8}

R: [4, 9, 4, 7, 7, 8, 1, 6, 1]    B: [2, 4, 5, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [689, 212, 43, 602, 172, 258, 645, 344, 159] . FixedPtCheck, [689, 212, 43, 602, 172, 258, 645, 344, 159]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{6, 8}, {1, 4, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 5 y2 - y3 + 5 y4 - y5, 0, 0, y1, 0, y2, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-16 y1 - 5 y2 + 33 y3 - 16 y5, 5 y1, -7 y1 + 16 y3 - 5 y4 - 7 y5, 5 y2, 5 y3, 0, 5 y4, 5 y5, 0]

 

  p = s 2 - s 6   p' = s 2 - s 6

 » SYNC'D 33201/1048576 , 0.03166294098

 
55 . Coloring, {2, 6, 9}

R: [4, 9, 4, 7, 7, 8, 1, 1, 2]    B: [2, 4, 5, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 18` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 `]`

For τ=1/2, [-2079, -1188, -125, -1950, -500, -150, -1875, -600, -891] . FixedPtCheck, [2079, 1188, 125, 1950, 500, 150, 1875, 600, 891]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y5 - y4 - y2 + 5 y3, y1, 0, y5, 0, 0, y4, y2, y3]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   8

See Matrix
 

[y1 + y2 - y3 - y4 - y5 + y6 + y7, y1, y2, y3, y4, y5, y6, y7, 0]

 

  p = - s 7 + s 8

 » SYNC'D 188803/8388608 , 0.02250707150

 
56 . Coloring, {2, 7, 8}

R: [4, 9, 4, 7, 7, 7, 5, 6, 1]    B: [2, 4, 5, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-442, -136, -258, -559, -1032, -129, -1290, -172, -102] . FixedPtCheck, [442, 136, 258, 559, 1032, 129, 1290, 172, 102]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, y2, y3, y5, y4, 0, y5]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y3, 3 y2 + 2 y1 - y4, y2, -y3 + 2 y2 + 3 y1, y1, 0, 0, y4, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 5709/131072 , 0.04355621338

 
57 . Coloring, {2, 7, 9}

R: [4, 9, 4, 7, 7, 7, 5, 1, 2]    B: [2, 4, 5, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + 3τ ` )` , 54` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + 3τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 5 + 3τ ` )``]`

For τ=1/2, [294, 168, 150, 375, 600, 25, 750, 100, 126] . FixedPtCheck, [294, 168, 150, 375, 600, 25, 750, 100, 126]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 4 vs 6 5 vs 7

Omega Rank for R :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - y2 + 3 y4, y1, 0, 3 y1 - y3 + 2 y4, y3, 0, y2, 0, y4]

 

  p = - s 3 + s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y3 - y2 + 2 y1 - y4, 2 y3 + 3 y1 - y5, y3, y2, y1, y4, 0, y5, 0]

 

  p' = s 4 - s 6   p = s 4 - s 6

 » SYNC'D 1301/131072 , 0.009925842285

 
58 . Coloring, {2, 8, 9}

R: [4, 9, 4, 7, 7, 7, 1, 6, 2]    B: [2, 4, 5, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-343, -196, -25, -325, -100, -75, -375, -100, -147] . FixedPtCheck, [343, 196, 25, 325, 100, 75, 375, 100, 147]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, y4, 0, y2, 0, -y3 + 5 y4 - y2 - y1 + 5 y5, y1, 0, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, 3 y1 - 4 y3 + 3 y2 - y4, y3, y2, 2 y1 - 3 y3 + 2 y2, 0, 0, y4, 0]

 

  p' = - s + s 5   p = - s + s 5

 » SYNC'D 15237/524288 , 0.02906227112

 
59 . Coloring, {3, 4, 5}

R: [4, 4, 5, 8, 3, 7, 1, 1, 1]    B: [2, 9, 4, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-15, -4, -3, -15, -4, -3, -7, -12, -1] . FixedPtCheck, [15, 4, 3, 15, 4, 3, 7, 12, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 3 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, 0, y4, y2, y3, 0, y1, -y5 + 5 y4 - y2 + 5 y3 - y1, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 7}, {6, 8}, {2, 9}},   net cycles: 2 .    order:   2

See Matrix
 

[0, 2 y2, 0, -y1 - 3 y2 + 2 y3, y1, y2, y3, -2 y2 + y3, -4 y2 + 2 y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 4509/2097152 , 0.002150058746

 
60 . Coloring, {3, 4, 6}

R: [4, 4, 5, 8, 7, 8, 1, 1, 1]    B: [2, 9, 4, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-1290, -344, -38, -1235, -152, -285, -494, -1140, -86] . FixedPtCheck, [1290, 344, 38, 1235, 152, 285, 494, 1140, 86]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 5 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

See Matrix
 

[y3, 0, 0, y3 - y2, y2, 0, y3 - y1, y1, 0]

 

  p' = - s 3 + s 4   p = s 3 - s 4

Omega Rank for B :  cycles: {{3, 4, 5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y5, y4, y3, y2, y4 - y3 - y2 + y1, y1, 0, -y5 + y4 + y1]

 

  p = s 2 - s 6   p' = s 2 - s 6

 » SYNC'D 2365/262144 , 0.009021759033

 
61 . Coloring, {3, 4, 7}

R: [4, 4, 5, 8, 7, 7, 5, 1, 1]    B: [2, 9, 4, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [555, 148, 171, 570, 684, 114, 741, 456, 37] . FixedPtCheck, [555, 148, 171, 570, 684, 114, 741, 456, 37]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 6 vs 8

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[7 y1, 0, 0, -7 y1 + 11 y3 + 11 y4 - 7 y2, 7 y3, 0, 7 y4, 7 y2, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{6, 8}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y2 + y3 + 4 y5 - y6, -y1 + 4 y3 - y4 + y5, y1, y2, 0, y3, y4, y5, y6]

 

  p = - s 5 + s 7   p' = - s 5 + s 7

 » SYNC'D 81607/4194304 , 0.01945662498

 
62 . Coloring, {3, 4, 8}

R: [4, 4, 5, 8, 7, 7, 1, 6, 1]    B: [2, 9, 4, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-255, -68, -19, -247, -76, -171, -247, -228, -17] . FixedPtCheck, [255, 68, 19, 247, 76, 171, 247, 228, 17]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 8

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: 0 .    order:   5

[y5, 0, 0, y4, y2, y3, y1, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {3, 4, 5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, y4, y2, y3, y5 - y3 + y6, 0, y1, -y4 + y2 + y1, y6 ]

 

  p = s 3 - s 7   p' = s 3 - s 7

 » SYNC'D 155757/16777216 , 0.009283840656

 
63 . Coloring, {3, 4, 9}

R: [4, 4, 5, 8, 7, 7, 1, 1, 2]    B: [2, 9, 4, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-429, -132, -17, -425, -68, -85, -221, -340, -33] . FixedPtCheck, [429, 132, 17, 425, 68, 85, 221, 340, 33]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 9 9 vs 9 3 vs 6 6 vs 9

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y1, y2, 0, y1 + y2 - y3, y3, 0]

 

  p = s 3 - s 5   p' = s 3 - s 4   p' = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 4, 5, 7}, {6, 8}},   net cycles: 3 .   

See Matrix
 

[y1, -y1 - 2 y3 + 2 y6 + 2 y5 - y2, -3 y3 + 2 y6 + 2 y5 - y4, y6, y5, -2 y3 + y6 + y5, y4, y3, y2]

 

  p' = - 1 - s 2 + s 6 + s 8   p' = 1 - s 3 - s 4 + s 7   p' = - 1 - s - s 2 + s 4 + s 5 + s 6

 » SYNC'D 95739/16777216 , 0.005706489086

 
64 . Coloring, {3, 5, 6}

R: [4, 4, 5, 7, 3, 8, 1, 1, 1]    B: [2, 9, 4, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-705, -188, -285, -741, -380, -57, -665, -228, -47] . FixedPtCheck, [705, 188, 285, 741, 380, 57, 665, 228, 47]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, 0, y4, y2, y3, 0, -y5 + 5 y4 - y2 + 5 y3 - y1, y1, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y3 + y4 - y5, 0, y3 + y4 - y1 - y2, y1, y2, y3, y4, y5]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

 » SYNC'D 4329/524288 , 0.008256912231

 
65 . Coloring, {3, 5, 7}

R: [4, 4, 5, 7, 3, 7, 5, 1, 1]    B: [2, 9, 4, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 18` (` 5 + τ ` )`` (` - 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 + τ ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [-165, -44, -513, -285, -684, -19, -399, -76, -11] . FixedPtCheck, [165, 44, 513, 285, 684, 19, 399, 76, 11]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y6, y5, y4, y3, y2, y1, -y6 - y3 - 3 y1 - 2 y4 + y2, y6 + y4 + y3 + 3 y1 - y2, -y6 - y5 - y3 - y2 - y1]
  p = s 2 - 2s 4 + 8s 5 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 4 vs 7

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   4

See Matrix
 

[-y1 + y2 + y3 - y4, 0, y1, y2, y3, 0, y4, 0, 0]

 

  p = s 4 - s 5

Omega Rank for B :  cycles: {{2, 9}, {6, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[2 y1, 9 y1 - 15 y2 - 11 y3 + 9 y4, 0, 2 y2, 0, 2 y3, 4 y2, 7 y1 - 9 y2 - 9 y3 + 7 y4, 2 y4]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 139/16384 , 0.008483886719

 
66 . Coloring, {3, 5, 8}

R: [4, 4, 5, 7, 3, 7, 1, 6, 1]    B: [2, 9, 4, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -27` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [-690, -184, -342, -741, -456, -171, -798, -228, -46] . FixedPtCheck, [690, 184, 342, 741, 456, 171, 798, 228, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y4 + 5 y3 - y2 - y5, 0, y1, y4, y3, y2, y5, 0, 0]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y3, y4, 0, y5, -5 y3 - 5 y5 + 14 y1 - 5 y2, 0, y1, -14 y3 - y4 - 14 y5 + 39 y1 - 14 y2, y2]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

 » SYNC'D 39/2048 , 0.01904296875

 
67 . Coloring, {3, 5, 9}

R: [4, 4, 5, 7, 3, 7, 1, 1, 2]    B: [2, 9, 4, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ ` )` , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ ` )``]`

For τ=1/2, [-234, -72, -102, -255, -136, -17, -238, -68, -18] . FixedPtCheck, [234, 72, 102, 255, 136, 17, 238, 68, 18]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, -y2 + 5 y1 - y5 + 5 y4 - y3, y1, y5, y4, 0, y3, 0, 0]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {5, 7}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y1, -y1 + 6 y2 + 6 y3 - 6 y4 - y5, 0, y2, 5 y2 + 5 y3 - 6 y4, y3, y4, 6 y2 + 6 y3 - 7 y4, y5]

 

  p' = s 2 + s 3 - s 5 - s 6   p = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 8

 » SYNC'D 16237/4194304 , 0.003871202469

 
68 . Coloring, {3, 6, 7}

R: [4, 4, 5, 7, 7, 8, 5, 1, 1]    B: [2, 9, 4, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , -9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [1335, 356, 855, 1482, 3420, 114, 3705, 456, 89] . FixedPtCheck, [1335, 356, 855, 1482, 3420, 114, 3705, 456, 89]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 5 8 vs 8

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y1, 0, 0, y2, y3, 0, y4, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   8

[y1, y2, y3, y4, 0, y5, y6, y7, y8]  

See Matrices
 

 » SYNC'D 318899/8388608 , 0.03801572323

 
69 . Coloring, {3, 6, 8}

R: [4, 4, 5, 7, 7, 8, 1, 6, 1]    B: [2, 9, 4, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-2055, -548, -171, -1995, -684, -855, -2223, -1140, -137] . FixedPtCheck, [2055, 548, 171, 1995, 684, 855, 2223, 1140, 137]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 - y2 + 5 y3 - y4 + 5 y5, 0, 0, y1, y2, y3, y4, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   8

[y4, y1, y2, y3, y5, 0, y6, y7, y8]  

See Matrices
 

 » SYNC'D 404463/16777216 , 0.02410787344

 
70 . Coloring, {3, 6, 9}

R: [4, 4, 5, 7, 7, 8, 1, 1, 2]    B: [2, 9, 4, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `3` (` 3 + τ 2 ` )` , -6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )` 2 , 3` (` 3 + τ 2 ` )` , -6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )` 2 , 3` (` 3 + τ 2 ` )` , -6` (` - 1 + τ ` )` , 3` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [13, 4, 1, 13, 4, 1, 13, 4, 1] . FixedPtCheck, [13, 4, 1, 13, 4, 1, 13, 4, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 3

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y2 - y1, y2, y1, -y2 - y1, y2, y1, -y2 - y1, y2, y1]
  p' = s 3 + s 4 + 4s 6   p' = s + s 4 - 4s 6   p' = s 2 + 3s 4 + 4s 6   p = s - 5s 5 - 12s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = - s 4 + s 5 - 2s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 7 2 vs 9 3 vs 9 2 vs 6 3 vs 9

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, y2, 0, y1, y2, 0, y1, y2, 0]

 

  p = s 2 - s 3   p' = - s 2 + s 4   p' = - s 2 + s 5   p' = - s 2 + s 3

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 4, 5, 6, 7, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y1, y2, y3, y1, y2, y3, y1, y2, y3]

 

  p' = - s + s 4   p' = - s 2 + s 5   p' = - s + s 7   p' = - s 2 + s 8   p' = - 1 + s 3   p' = - 1 + s 6


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x3, x4, x2, -5 x1 + x3 + 3 x4 + x2 - x5, x5, -12 x1 - 4 x3 + 4 x4 + 4 x2 - 2 x5]
For A+2Δ :   [y1, -5 y1 - 5 y2 - y3 - 5 y5 - y6, 7 y1 + 7 y2 + 7 y5 - y4 - y7, y2, y3, y4, y5, y6, y7]
For A-2Δ :   [-y3 - y2, -y4 - y5, -y1 - y6, y3, y4, y1, y2, y5, y6]

Range of {ΩΔi}: [-μ1 - μ2, μ1, μ2, -μ1 - μ2, μ1, μ2, -μ1 - μ2, μ1, μ2]

 
rank of M is 9 , rank of N is 7

M              \ ;    N

$ [ [0, 0, 0, 3, 0, 0, 3, 0, 0] , [0, 0, 0, 0, 2, 0, 0, 2, 0] , [0, 0, 0, 0, 0, 1, 0, 0, 1] , [3, 0, 0, 0, 0, 0, 3, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 2, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 1] , [3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 2, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 1, 0, 0, 0] ] $     $ [ [0, 186, 280, 535, 414, 480, 535, 470, 310] , [186, 0, 293, 434, 535, 405, 450, 535, 372] , [280, 293, 0, 520, 525, 535, 270, 252, 535] , [535, 434, 520, 0, 126, 90, 535, 510, 460] , [414, 535, 525, 126, 0, 180, 530, 535, 365] , [480, 405, 535, 90, 180, 0, 500, 485, 535] , [535, 450, 270, 535, 530, 500, 0, 90, 300] , [470, 535, 252, 510, 535, 485, 90, 0, 333] , [310, 372, 535, 460, 365, 535, 300, 333, 0] ] $

Check is ΩΔN zero? true, πΔ= [2, -1, -1, 2, -1, -1, 2, -1, -1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x1, x2, x3, x4, x5, x6, x7, x8, x9]

τ= 27 , r'= 2/3

Ranges

Action of R on ranges, [[1], [1], [2]]
Action of B on ranges, [[2], [3], [1]]
β({1, 4, 7}) = 1/2
β({2, 5, 8}) = 1/3
β({3, 6, 9}) = 1/6

ker N, [-μ2 - μ1, μ1, μ2, -μ2 - μ1, μ1, μ2, -μ2 - μ1, μ1, μ2]
Range of N
    [y4, y3, y1, y2, y4 - y3 + y2 + y6 - y7, y4 - y1 + y2 + y6 - y5, y6, y7, y5]

Partitions

Too many possibilities to consider

Sandwich
Coloring {3, 6, 9}
Rank3
R,B [4, 4, 5, 7, 7, 8, 1, 1, 2], [2, 9, 4, 8, 3, 7, 5, 6, 1]
π2 [0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0]
u2 [186, 280, 535, 414, 480, 535, 470, 310, 293, 434, 535, 405, 450, 535, 372, 520, 525, 535, 270, 252, 535, 126, 90, 535, 510, 460, 180, 530, 535, 365, 500, 485, 535, 90, 300, 333] (dim 1)
wpp [3, 3, 3, 3, 3, 3, 3, 3, 3]
π3 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u3 [323, 255, 195, 235, 303, 363, 0, 795, 447, 675, 45, 70, 165, 15, 105, 1605, 1335, 705, 110, 1227, 1047, 685, 1335, 1095, 765, 75, 225, 245, 531, 849, 489, 25, 30, 390, 75, 125, 1047, 1227, 646, 150, 1335, 1605, 606, 855, 1065, 726, 15, 470, 510, 343, 225, 765, 686, 1335, 510, 785, 726, 1065, 705, 606, 1605, 0, 105, 150, 0, 363, 303, 35, 165, 145, 45, 195, 675, 699, 430, 390, 30, 255, 375, 489, 215, 795, 849, 55]

 

 
71 . Coloring, {3, 7, 8}

R: [4, 4, 5, 7, 7, 7, 5, 6, 1]    B: [2, 9, 4, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-645, -172, -513, -741, -2052, -171, -2223, -228, -43] . FixedPtCheck, [645, 172, 513, 741, 2052, 171, 2223, 228, 43]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y6, y5, y4, y3, y2, y1, -y6 - y3 + y2 + y1, -y4 + y6 + y3 - y2 - y1, -y6 - y5 - y3 - y2 - y1]
  p = s 3 + s 4 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 6 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, y3, y4, 2 y1, y2, 0, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   6

[y2, y3, y4, y1, 0, 0, 0, y6, y5]  

See Matrices
 

 » SYNC'D 59/512 , 0.1152343750

 
72 . Coloring, {3, 7, 9}

R: [4, 4, 5, 7, 7, 7, 5, 1, 2]    B: [2, 9, 4, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-13, -4, -9, -15, -36, -1, -39, -4, -1] . FixedPtCheck, [13, 4, 9, 15, 36, 1, 39, 4, 1]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 5 6 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y1, y1, 0, y3, y2, 0, -3 y1 + y3 + y2, 0, 0]

 

  p = - s 3 + s 4   p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y6, y5, y4, y3, 0, y1, 0, y2, -y6 - y5 + y4 + y3 + y1 + y2]

 

  p = - s 3 - s 4 + s 6 + s 7

 » SYNC'D 981/32768 , 0.02993774414

 
73 . Coloring, {3, 8, 9}

R: [4, 4, 5, 7, 7, 7, 1, 6, 2]    B: [2, 9, 4, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ 2 ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , -54` (` - 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [3302, 1016, 306, 3315, 1224, 765, 3978, 1020, 254] . FixedPtCheck, [3302, 1016, 306, 3315, 1224, 765, 3978, 1020, 254]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y2, y3, 0, y1, y3, 2 y3, y4, 0, 0]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y3, y1, y2, y7, y6, 0, 0, y4, y5]  

See Matrices
 

 » SYNC'D 6075/131072 , 0.04634857178

 
74 . Coloring, {4, 5, 6}

R: [4, 4, 4, 8, 3, 8, 1, 1, 1]    B: [2, 9, 5, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` - 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-735, -196, -57, -741, -76, -171, -247, -684, -49] . FixedPtCheck, [735, 196, 57, 741, 76, 171, 247, 684, 49]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y3 - y1 - 3 y2 - 3 y6, -y3 + 2 y2 + 2 y6 - y5, -y3 - y4, y1, y2, y6, y3, y4, y5]
  p = s 3 + s 4 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 4 3 vs 5

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

[y1, 0, y4, y3, 0, 0, 0, y2, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   2

See Matrix
 

[0, y2 - y3, 0, 0, y2 - y1, y1, y2, 0, y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4

 » SYNC'D 51/2048 , 0.02490234375

 
75 . Coloring, {4, 5, 7}

R: [4, 4, 4, 8, 3, 7, 5, 1, 1]    B: [2, 9, 5, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [1095, 292, 513, 1425, 684, 285, 741, 1140, 73] . FixedPtCheck, [1095, 292, 513, 1425, 684, 285, 741, 1140, 73]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   6

[y1, 0, y4, y5, y3, 0, y2, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y2 + 4 y4 - y1 - y3, 4 y2 - y5 + y4, 0, 0, y1, y2, y5, y4, y3]

 

  p' = s 4 - s 6   p = s 4 - s 6

 » SYNC'D 138339/4194304 , 0.03298258781

 
76 . Coloring, {4, 5, 8}

R: [4, 4, 4, 8, 3, 7, 1, 6, 1]    B: [2, 9, 5, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [-1335, -356, -285, -1482, -380, -1026, -1235, -1368, -89] . FixedPtCheck, [1335, 356, 285, 1482, 380, 1026, 1235, 1368, 89]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: 0 .    order:   5

[y3, 0, y1, y2, 0, y4, y5, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y4 - y1, y3 - y2, 0, 0, y4, 0, y3, y2, y1]

 

  p' = - s 3 + s 5   p = s 3 - s 5

 » SYNC'D 351/16384 , 0.02142333984

 
77 . Coloring, {4, 5, 9}

R: [4, 4, 4, 8, 3, 7, 1, 1, 2]    B: [2, 9, 5, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-481, -148, -51, -510, -68, -102, -221, -408, -37] . FixedPtCheck, [481, 148, 51, 510, 68, 102, 221, 408, 37]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -2 .    order:   3

See Matrix
 

[y2, y3, 2 y3, y1, 0, 0, y3, y4, 0]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{6, 8}, {1, 2, 9}, {5, 7}},   net cycles: 3 .    order:   6

See Matrix
 

[-y1 + 2 y2 + 2 y3 - y4, y1, 0, 0, y2 + 2 y3, y2, 2 y2 + y3, y3, y4]

 

  p = s - s 3 - s 4 + s 6   p = - s + s 7   p = - s - s 2 + s 4 + s 5

 » SYNC'D 15525/1048576 , 0.01480579376

 
78 . Coloring, {4, 6, 7}

R: [4, 4, 4, 8, 7, 8, 5, 1, 1]    B: [2, 9, 5, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-690, -184, -114, -741, -456, -171, -570, -684, -46] . FixedPtCheck, [690, 184, 114, 741, 456, 171, 570, 684, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[-5 y4 + 13 y3 + 13 y2 - 5 y1, 0, 0, 5 y4, 5 y3, 0, 5 y2, 5 y1, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 + 3 y2 - y3 - y5, 3 y1 + 2 y2 - y4, y1, 0, y2, y3, y4, 0, y5]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

 » SYNC'D 7031/524288 , 0.01341056824

 
79 . Coloring, {4, 6, 8}

R: [4, 4, 4, 8, 7, 8, 1, 6, 1]    B: [2, 9, 5, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )` 2 ` (` - 5 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-405, -108, -19, -399, -76, -513, -285, -684, -27] . FixedPtCheck, [405, 108, 19, 399, 76, 513, 285, 684, 27]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y4 - 2 y5 - y1 - 3 y2 + y3, y4 + 2 y5 + 2 y2 - 2 y3 - y6, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 2 - 2s 4 + 8s 5 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 5 3 vs 6

Omega Rank for R :  cycles: {{6, 8}},   net cycles: 0 .    order:   4

[y4, 0, 0, y3, 0, y1, y2, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 0 .    order:   2

See Matrix
 

[2 y3, 2 y2, 2 y1, 0, -4 y3 + 5 y2 - 2 y1, 0, 4 y3, 0, -2 y3 + 2 y2]

 

  p' = s 3 - s 5   p = s 2 - s 6   p' = s 2 - s 4

 » SYNC'D 5/512 , 0.009765625000

 
80 . Coloring, {4, 6, 9}

R: [4, 4, 4, 8, 7, 8, 1, 1, 2]    B: [2, 9, 5, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [39, 12, 1, 39, 4, 9, 15, 36, 3] . FixedPtCheck, [39, 12, 1, 39, 4, 9, 15, 36, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 2 vs 5 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y1 - y2, 0, 2 y1 - y2, 0, 0, 2 y1 - 2 y2, y2, 0]

 

  p = s 2 - s 5   p' = s 2 - s 4   p' = s 3 - s 4

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 5}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + y2 + y3 - y4, y1, y2 + y3 - y5, 0, y2, y3, y5, 0, y4]

 

  p = - s 3 + s 6   p' = - s 3 + s 6

 » SYNC'D 7695/524288 , 0.01467704773

 
81 . Coloring, {4, 7, 8}

R: [4, 4, 4, 8, 7, 7, 5, 6, 1]    B: [2, 9, 5, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [555, 148, 285, 741, 1140, 513, 1425, 684, 37] . FixedPtCheck, [555, 148, 285, 741, 1140, 513, 1425, 684, 37]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 4 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y3, 0, 0, y2, y1, y5, y6, y4, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}, {3, 5}},   net cycles: 0 .    order:   4

See Matrix
 

[2 y1 + 3 y2 - y4, 3 y1 + 2 y2 - 4 y3, y1, 0, y2, 0, 3 y3, y3, y4]

 

  p' = s 4 - s 6   p' = s 3 - s 5   p = s 3 - s 7

 » SYNC'D 213/8192 , 0.02600097656

 
82 . Coloring, {4, 7, 9}

R: [4, 4, 4, 8, 7, 7, 5, 1, 2]    B: [2, 9, 5, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-13, -4, -3, -15, -12, -3, -15, -12, -1] . FixedPtCheck, [13, 4, 3, 15, 12, 3, 15, 12, 1]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 8 8 vs 8 4 vs 6 5 vs 8

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y1, 0, -y2 - y1 + 4 y4 - y3, y4, 0, y4, y3, 0]

 

  p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 5}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[-y1 + 4 y5 + 4 y2 - y4 - y3, y1, y5, 0, y2, y5, y4, y2, y3]

 

  p = s 2 - s 8   p' = s 2 - s 4 - s 5 + s 7   p' = s 3 + s 4 - s 6 - s 7

 » SYNC'D 210735/16777216 , 0.01256078482

 
83 . Coloring, {4, 8, 9}

R: [4, 4, 4, 8, 7, 7, 1, 6, 2]    B: [2, 9, 5, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-1287, -396, -85, -1326, -340, -918, -1275, -1224, -99] . FixedPtCheck, [1287, 396, 85, 1326, 340, 918, 1275, 1224, 99]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 3 vs 7

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: 0 .    order:   5

[y2, y1, 0, y4, 0, y3, y6, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[y3 + y2, y3 + y2, y1, 0, 3 y2 - y1, 0, 3 y3, y3, y2]

 

  p = s 2 - s 6   p' = s 2 - s 6   p' = s 3 - s 5   p' = s 4 - s 6

 » SYNC'D 90255/4194304 , 0.02151846886

 
84 . Coloring, {5, 6, 7}

R: [4, 4, 4, 7, 3, 8, 5, 1, 1]    B: [2, 9, 5, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-375, -100, -513, -741, -684, -57, -741, -228, -25] . FixedPtCheck, [375, 100, 513, 741, 684, 57, 741, 228, 25]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

[y1, 0, y3, y4, y2, 0, y5, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y6, y5, 0, 0, y3, y4, y2, 3 y3, y1]

 

  p = s 5 - s 7

 » SYNC'D 10359/524288 , 0.01975822449

 
85 . Coloring, {5, 6, 8}

R: [4, 4, 4, 7, 3, 8, 1, 6, 1]    B: [2, 9, 5, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-705, -188, -171, -798, -228, -342, -741, -456, -47] . FixedPtCheck, [705, 188, 171, 798, 228, 342, 741, 456, 47]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 - y2 + 5 y3 - y4 + 5 y5, 0, y1, y2, 0, y3, y4, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[5 y1, -16 y1 + 33 y4 - 5 y3 - 16 y2, 0, 0, -7 y1 + 16 y4 - 7 y2, 0, 5 y4, 5 y3, 5 y2]

 

  p' = s 3 - s 5   p = s 3 - s 5

 » SYNC'D 2439/131072 , 0.01860809326

 
86 . Coloring, {5, 6, 9}

R: [4, 4, 4, 7, 3, 8, 1, 1, 2]    B: [2, 9, 5, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [1157, 356, 255, 1326, 340, 102, 1105, 408, 89] . FixedPtCheck, [1157, 356, 255, 1326, 340, 102, 1105, 408, 89]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, y4, 2 y4, y2, 0, 0, y3, y4, 0]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + y3 + y4 - y5, y1, 0, 0, y3 + y4 - y2, y2, y3, y4, y5]

 

  p = - s 3 + s 6   p' = - s 3 + s 6

 » SYNC'D 915/32768 , 0.02792358398

 
87 . Coloring, {5, 7, 8}

R: [4, 4, 4, 7, 3, 7, 5, 6, 1]    B: [2, 9, 5, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 4 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-1470, -392, -3078, -3705, -4104, -855, -4446, -1140, -98] . FixedPtCheck, [1470, 392, 3078, 3705, 4104, 855, 4446, 1140, 98]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y5, 0, y1, y2, y3, 2 y5, y4, 0, 0]

 

  p = - s 2 + s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y3, 0, 0, y2, 0, y4, 4 y2, y5]

 

  p = s 4 - s 6

 » SYNC'D 299/4096 , 0.07299804688

 
88 . Coloring, {5, 7, 9}

R: [4, 4, 4, 7, 3, 7, 5, 1, 2]    B: [2, 9, 5, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 3τ 2 ` )` , -54` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )` 3 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [-598, -184, -918, -1275, -1224, -85, -1326, -340, -46] . FixedPtCheck, [598, 184, 918, 1275, 1224, 85, 1326, 340, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 4 vs 6 4 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y1, y1, y3, 3 y1 + y3 - y2 + y4, y2, 0, y4, 0, 0]

 

  p = - s 2 + s 6   p = - s 2 + s 3 - s 4 + s 5

Omega Rank for B :  cycles: {{6, 8}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y3 + 2 y4, 2 y3, 0, 0, 2 y4, 2 y2, -2 y1 + 2 y3 + 2 y4, 3 y3 + 3 y4 - 2 y2, 2 y1]

 

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 903/65536 , 0.01377868652

 
89 . Coloring, {5, 8, 9}

R: [4, 4, 4, 7, 3, 7, 1, 6, 2]    B: [2, 9, 5, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [559, 172, 153, 663, 204, 153, 663, 204, 43] . FixedPtCheck, [559, 172, 153, 663, 204, 153, 663, 204, 43]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}},   net cycles: -2 .    order:   3

See Matrix
 

[y1, y2, 2 y2, y3, 0, 2 y2, y4, 0, 0]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y2 + 2 y1 + 2 y3 - y4 - y5, y2, 0, 0, y1, 0, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 3045/65536 , 0.04646301270

 
90 . Coloring, {6, 7, 8}

R: [4, 4, 4, 7, 7, 8, 5, 6, 1]    B: [2, 9, 5, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -18` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-15, -4, -9, -21, -36, -9, -45, -12, -1] . FixedPtCheck, [15, 4, 9, 21, 36, 9, 45, 12, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 4 vs 7

Omega Rank for R :  cycles: {{6, 8}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y3 + 4 y1 - y2, 0, 0, -y4 + 4 y3 + y1, y4, y3, y2, y1, 0]

 

  p' = s 3 - s 5   p = s 3 - s 5

Omega Rank for B :  cycles: {{3, 5}, {2, 9}},   net cycles: 0 .    order:   4

See Matrix
 

[2 y1 + 3 y2 - y4, 3 y1 + 2 y2 - 4 y3, y1, 0, y2, 0, y3, 3 y3, y4]

 

  p' = - s 4 + s 6   p = s 3 - s 5   p' = - s 3 + s 5

 » SYNC'D 2865/262144 , 0.01092910767

 
91 . Coloring, {6, 7, 9}

R: [4, 4, 4, 7, 7, 8, 5, 1, 2]    B: [2, 9, 5, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-481, -148, -255, -663, -1020, -51, -1275, -204, -37] . FixedPtCheck, [481, 148, 255, 663, 1020, 51, 1275, 204, 37]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 8 8 vs 8 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y3, y5, 0, y1, y2, 0, y4, y5, 0]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y2 + 5 y3 + 5 y1 - y4 - y5 - y6 - y7, y2, y3, 0, y1, y4, y5, y6, y7]

 

  p = - s 4 - s 5 + s 7 + s 8

 » SYNC'D 285713/8388608 , 0.03405964375

 
92 . Coloring, {6, 8, 9}

R: [4, 4, 4, 7, 7, 8, 1, 6, 2]    B: [2, 9, 5, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` - 5 - 3τ - τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 - 3τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-689, -212, -51, -714, -204, -306, -765, -408, -53] . FixedPtCheck, [689, 212, 51, 714, 204, 306, 765, 408, 53]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, y4, 0, y3, 0, y2, -y5 - y4 - y3 + 5 y2 + 5 y1, y1, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[7 y1, -7 y1 + 11 y2 + 11 y3 - 10 y5 - 7 y4, 7 y2, 0, 7 y3, 0, 7 y5, 21 y5, 7 y4]

 

  p = s 2 + s 3 - s 5 - s 6   p' = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 181071/4194304 , 0.04317069054

 
93 . Coloring, {7, 8, 9}

R: [4, 4, 4, 7, 7, 7, 5, 6, 2]    B: [2, 9, 5, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )` 2 , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-26, -8, -18, -39, -72, -9, -90, -12, -2] . FixedPtCheck, [26, 8, 18, 39, 72, 9, 90, 12, 2]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y4, 0, y3, y2, 2 y4, y1, 0, 0]

 

  p = s 3 - s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, y4, y3, 0, y2, 0, 0, y1, -y5 - y4 + 5 y3 + 5 y2 - y1]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 59/512 , 0.1152343750

 
94 . Coloring, {2, 3, 4, 5}

R: [4, 9, 5, 8, 3, 7, 1, 1, 1]    B: [2, 4, 4, 7, 7, 8, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-741, -228, -129, -645, -172, -129, -301, -516, -171] . FixedPtCheck, [741, 228, 129, 645, 172, 129, 301, 516, 171]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y5, 0, y2, y3, y4, 0, y1, -y5 + 5 y2 - y3 + 5 y4 - 3 y1, 2 y1]

 

  p = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, 4 y2 - y3 + y4, 0, -y1 + y2 + 4 y4, y1, y2, y3, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

 » SYNC'D 10567/524288 , 0.02015495300

 
95 . Coloring, {2, 3, 4, 6}

R: [4, 9, 5, 8, 7, 8, 1, 1, 1]    B: [2, 4, 4, 7, 3, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [78, 24, 2, 65, 8, 15, 26, 60, 18] . FixedPtCheck, [78, 24, 2, 65, 8, 15, 26, 60, 18]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, 0, y2, y3, 0, y4, y5, 2 y3]

 

  p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 2 y1, y3, y4, y5, y1, y2, 0, 0]

 

  p = s 2 - s 6

 » SYNC'D 945/32768 , 0.02883911133

 
96 . Coloring, {2, 3, 4, 7}

R: [4, 9, 5, 8, 7, 7, 5, 1, 1]    B: [2, 4, 4, 7, 3, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [481, 148, 129, 430, 516, 86, 559, 344, 111] . FixedPtCheck, [481, 148, 129, 430, 516, 86, 559, 344, 111]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-7 y4 + 11 y3 + 11 y2 - 7 y1 - 7 y5, 0, 0, 7 y4, 7 y3, 0, 7 y2, 7 y1, 7 y5]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-y2 + y5 + 4 y4, 4 y5 + y4 - y1 - y3, y1, y2, 0, y5, y3, y4, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 48475/2097152 , 0.02311468124

 
97 . Coloring, {2, 3, 4, 8}

R: [4, 9, 5, 8, 7, 7, 1, 6, 1]    B: [2, 4, 4, 7, 3, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [663, 204, 43, 559, 172, 387, 559, 516, 153] . FixedPtCheck, [663, 204, 43, 559, 172, 387, 559, 516, 153]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: -1 .    order:   5

See Matrix
 

[y1, 0, 0, y2, y3, y4, y5, y6, 2 y3]

 

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

[y4, y3, y1, y2, y5, 0, y7, y6, 0]  

See Matrices
 

 » SYNC'D 306315/16777216 , 0.01825779676

 
98 . Coloring, {2, 3, 4, 9}

R: [4, 9, 5, 8, 7, 7, 1, 1, 2]    B: [2, 4, 4, 7, 3, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-693, -396, -25, -625, -100, -125, -325, -500, -297] . FixedPtCheck, [693, 396, 25, 625, 100, 125, 325, 500, 297]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y1, 0, -y2 + 5 y1 - y6 - y5 - y3 + 5 y4, y6, 0, y5, y3, y4]

 

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{6, 8}, {3, 4, 5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y3, y4, y5, y6, -y3 - y6 + 2 y1 + 3 y2, y1, -y4 - y5 + 3 y1 + 2 y2, y2, 0]

 

  p = - s 3 + s 7   p' = - s 3 + s 7

 » SYNC'D 101475/8388608 , 0.01209676266

 
99 . Coloring, {2, 3, 5, 6}

R: [4, 9, 5, 7, 3, 8, 1, 1, 1]    B: [2, 4, 4, 8, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [1833, 564, 645, 1677, 860, 129, 1505, 516, 423] . FixedPtCheck, [1833, 564, 645, 1677, 860, 129, 1505, 516, 423]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[5 y1 - y2 + 5 y3 - y4 - 3 y5, 0, y1, y2, y3, 0, y4, y5, 2 y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[0, y1, 0, y2, y3, y4, y5, y6, 0]  

See Matrices
 

 » SYNC'D 9411/262144 , 0.03590011597

 
100 . Coloring, {2, 3, 5, 7}

R: [4, 9, 5, 7, 3, 7, 5, 1, 1]    B: [2, 4, 4, 8, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , -18` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [429, 132, 1161, 645, 1548, 43, 903, 172, 99] . FixedPtCheck, [429, 132, 1161, 645, 1548, 43, 903, 172, 99]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y6 - y4 + y5 - 3 y3 + y2 - 2 y1, y6 - 2 y5 + 2 y3 - y2 + y1, y6, y4, y5, y3, -y6 - y2, y2, y1]
  p = s 3 + 3s 4 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[y4, 0, y3, y2, y1, 0, -y4 - y3 + y2 + y1 + y5, 0, y5]

 

  p = s 5 - s 6

Omega Rank for B :  cycles: {{6, 8}},   net cycles: 0 .    order:   6

[y1, y2, 0, y3, 0, y4, y5, y6, 0]  

See Matrices
 

 » SYNC'D 4663/131072 , 0.03557586670

 
101 . Coloring, {2, 3, 5, 8}

R: [4, 9, 5, 7, 3, 7, 1, 6, 1]    B: [2, 4, 4, 8, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 3 + τ 2 ` )` , -54` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [598, 184, 258, 559, 344, 129, 602, 172, 138] . FixedPtCheck, [598, 184, 258, 559, 344, 129, 602, 172, 138]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[y2, 0, y1, -y2 + 5 y1 + 5 y5 - y3 - 2 y4, y5, y4, y3, 0, y4]

 

  p = - s 2 + s 4 + s 5 - s 7   p' = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y4, y3, 0, -y4 - 14 y3 + 39 y2 - 14 y1, y2, 0, -5 y3 + 14 y2 - 5 y1, y1, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 4653/524288 , 0.008874893188

 
102 . Coloring, {2, 3, 5, 9}

R: [4, 9, 5, 7, 3, 7, 1, 1, 2]    B: [2, 4, 4, 8, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 + τ ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ ` )`` (` 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-378, -216, -150, -375, -200, -25, -350, -100, -162] . FixedPtCheck, [378, 216, 150, 375, 200, 25, 350, 100, 162]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y5 - y2 - y3 - y1 - y6, y5, y4, y2, y3, y1, -3 y5 - 2 y2 - y3 - 2 y1 - 4 y6 - y4, 3 y5 + 2 y2 + y3 + 2 y1 + 4 y6, y6]
  p = s 2 - 8s 4 - 12s 5 + 8s 6 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 8 vs 8 8 vs 8 4 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}, {3, 5}},   net cycles: 3 .    order:   6

See Matrix
 

[4 y3 + 4 y1 - y2 - y4, y3, y1, y2, y3, 0, y4, 0, y1]

 

  p = - s - s 2 + s 4 + s 5   p = - s + s 7   p = s - s 3 - s 4 + s 6

Omega Rank for B :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[4 y1, 9 y1 + 9 y2 + 9 y3 - 13 y4 - 4 y5, 0, 4 y2, 5 y1 + 5 y2 + 5 y3 - 9 y4, 4 y3, 4 y4, 4 y5, 0]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

 » SYNC'D 8073/4194304 , 0.001924753189

 
103 . Coloring, {2, 3, 6, 7}

R: [4, 9, 5, 7, 7, 8, 5, 1, 1]    B: [2, 4, 4, 8, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [1157, 356, 645, 1118, 2580, 86, 2795, 344, 267] . FixedPtCheck, [1157, 356, 645, 1118, 2580, 86, 2795, 344, 267]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, 0, y5, y4, 0, y3, y2, 2 y2]

 

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 0 .    order:   6

[y6, y7, y5, y4, 0, y3, y1, y2, 0]  

See Matrices
 

 » SYNC'D 97569/2097152 , 0.04652452469

 
104 . Coloring, {2, 3, 6, 8}

R: [4, 9, 5, 7, 7, 8, 1, 6, 1]    B: [2, 4, 4, 8, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [1781, 548, 129, 1505, 516, 645, 1677, 860, 411] . FixedPtCheck, [1781, 548, 129, 1505, 516, 645, 1677, 860, 411]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y2, 0, 0, -y2 - 3 y1 + 5 y5 - y4 + 5 y3, y1, y5, y4, y3, 2 y1]

 

  p' = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y6, y5, y4, y3, y2, 0, y1, y7, 0]  

See Matrices
 

 » SYNC'D 119025/4194304 , 0.02837777138

 
105 . Coloring, {2, 3, 6, 9}

R: [4, 9, 5, 7, 7, 8, 1, 1, 2]    B: [2, 4, 4, 8, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [357, 204, 25, 325, 100, 25, 325, 100, 153] . FixedPtCheck, [357, 204, 25, 325, 100, 25, 325, 100, 153]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[5 y1 - y2 - 2 y4 - y3 + 5 y5, y1, 0, y2, y4, 0, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y7, y6, y5, y4, y3, y2, y1, y7 - y6 - y5 + y4 + y3 + y2 - y1, 0]

 

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

 » SYNC'D 1366845/67108864 , 0.02036757767

 
106 . Coloring, {2, 3, 7, 8}

R: [4, 9, 5, 7, 7, 7, 5, 6, 1]    B: [2, 4, 4, 8, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )``]`

For τ=1/2, [13, 4, 9, 13, 36, 3, 39, 4, 3] . FixedPtCheck, [13, 4, 9, 13, 36, 3, 39, 4, 3]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y6, y5, y4, y3, y2, y1, y6 + 2 y5 + y3 + 3 y2 + 3 y1, -y6 - 2 y5 - y4 - y3 - 3 y2 - 3 y1, -y6 - y5 - y3 - y2 - y1]
  p = s 2 + 2s 4 + 8s 5 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 5

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, 0, 0, y1, y3, y5, y4, 0, y5]

 

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y1, y2, y3, y4, 0, 0, 0, y5, 0]  

See Matrices
 

 » SYNC'D 417/8192 , 0.05090332031

 
107 . Coloring, {2, 3, 7, 9}

R: [4, 9, 5, 7, 7, 7, 5, 1, 2]    B: [2, 4, 4, 8, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-357, -204, -225, -375, -900, -25, -975, -100, -153] . FixedPtCheck, [357, 204, 225, 375, 900, 25, 975, 100, 153]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{2, 9}, {5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - y3 + 3 y4, y1, 0, 3 y1 - y2 + 2 y4, y2, 0, y3, 0, y4]

 

  p' = - s 3 + s 5   p = - s 3 + s 5

Omega Rank for B :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y2, y3, y2, y3 - y2 - y1 + y4, 0, y1, 0, y4, 0]

 

  p = - s 4 + s 5   p = - s 4 + s 6

 » SYNC'D 9801/262144 , 0.03738784790

 
108 . Coloring, {2, 3, 8, 9}

R: [4, 9, 5, 7, 7, 7, 1, 6, 2]    B: [2, 4, 4, 8, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 54` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 1 + τ ` )`` (`5 + 2τ + 8τ 2 - 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [1778, 1016, 150, 1625, 600, 375, 1950, 500, 762] . FixedPtCheck, [1778, 1016, 150, 1625, 600, 375, 1950, 500, 762]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[5 y4 - y2 - 3 y3 - y1 + 5 y5, y4, 0, y2, y3, 2 y3, y1, 0, y5]

 

  p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y4, y3, y1, y2, -y4 + y3 + y1 - y2 + y5, 0, 0, y5, 0]

 

  p = - s 3 + s 4 - s 5 + s 6

 » SYNC'D 7235/262144 , 0.02759933472

 
109 . Coloring, {2, 4, 5, 6}

R: [4, 9, 4, 8, 3, 8, 1, 1, 1]    B: [2, 4, 5, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1911, 588, 129, 1677, 172, 387, 559, 1548, 441] . FixedPtCheck, [1911, 588, 129, 1677, 172, 387, 559, 1548, 441]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y1 - 3 y3 + y5 - 2 y6 - y2, 2 y1 + 2 y3 - y5 + y6, -y5 - y4, y2, y1, y3, y5, y4, y6]
  p = s 2 + 2s 4 + 8s 5 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 4 vs 5

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, 0, y4, y2, 0, 0, 0, y3, y4]

 

  p = s 2 - s 5

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 2 y4, 0, y2, y3, y4, y1, 0, 0]

 

  p = - s 3 + s 5

 » SYNC'D 9/128 , 0.07031250000

 
110 . Coloring, {2, 4, 5, 7}

R: [4, 9, 4, 8, 3, 7, 5, 1, 1]    B: [2, 4, 5, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-949, -292, -387, -1075, -516, -215, -559, -860, -219] . FixedPtCheck, [949, 292, 387, 1075, 516, 215, 559, 860, 219]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, 0, y2, y3, y4, 0, y6, y5, 2 y6]

 

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, y4, 0, y3, -y5 - y3 - 15 y2 + 4 y4 + 4 y1, y2, y1, y4 - 4 y2 + y1, 0]

 

  p' = - s 2 + s 6   p = - s 2 + s 6

 » SYNC'D 507/32768 , 0.01547241211

 
111 . Coloring, {2, 4, 5, 8}

R: [4, 9, 4, 8, 3, 7, 1, 6, 1]    B: [2, 4, 5, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [-3471, -1068, -645, -3354, -860, -2322, -2795, -3096, -801] . FixedPtCheck, [3471, 1068, 645, 3354, 860, 2322, 2795, 3096, 801]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 6

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: -1 .    order:   5

See Matrix
 

[y1, 0, y6, y2, 0, y3, y4, y5, y6]

 

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y1, y2, 0, y3, y4, 0, y5, y6, 0]  

See Matrices
 

 » SYNC'D 20385/524288 , 0.03888130188

 
112 . Coloring, {2, 4, 5, 9}

R: [4, 9, 4, 8, 3, 7, 1, 1, 2]    B: [2, 4, 5, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-777, -444, -75, -750, -100, -150, -325, -600, -333] . FixedPtCheck, [777, 444, 75, 750, 100, 150, 325, 600, 333]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2, y1, -y4 - y3, -y2 - 2 y1 + y5 + y3 + y6, y1 - 2 y5 - y3 - 2 y6, y5, y4, y3, y6]
  p = s 2 - 4s 5 - 8s 6 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}, {2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[y3, y4, 2 y1, y2, 0, 0, y1, -y3 + 5 y4 - y2 - 3 y1 + 5 y5, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y4, y3, 0, y2, -y4 - y2 + 2 y1 + 3 y5, y1, -y3 + 3 y1 + 2 y5, y5, 0]

 

  p = - s 4 + s 6   p' = s 4 - s 6

 » SYNC'D 111537/8388608 , 0.01329624653

 
113 . Coloring, {2, 4, 6, 7}

R: [4, 9, 4, 8, 7, 8, 5, 1, 1]    B: [2, 4, 5, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ 2 ` )`` (` 5 + 3τ 2 ` )` , -54` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [598, 184, 86, 559, 344, 129, 430, 516, 138] . FixedPtCheck, [598, 184, 86, 559, 344, 129, 430, 516, 138]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-5 y1 + 13 y2 + 13 y3 - 5 y4 - 5 y5, 0, 0, 5 y1, 5 y2, 0, 5 y3, 5 y4, 5 y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y2 - y1 + 3 y3 - y4, 3 y2 + 2 y3 - y5, y2, y1, y3, y4, y5, 0, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 10669/1048576 , 0.01017475128

 
114 . Coloring, {2, 4, 6, 8}

R: [4, 9, 4, 8, 7, 8, 1, 6, 1]    B: [2, 4, 5, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -18` (` - 5 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , -9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )`` (` 1 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` - 5 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3 `]`

For τ=1/2, [1053, 324, 43, 903, 172, 1161, 645, 1548, 243] . FixedPtCheck, [1053, 324, 43, 903, 172, 1161, 645, 1548, 243]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y2 + y6 - 2 y5 - 2 y4 - y3 - y1, 2 y2 - 2 y6 + y5 + 2 y4 + y3, -y4 - y3, y1, y2, y6, y3, y4, y5]
  p = s 3 + 3s 4 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 6 vs 6

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, 0, 0, y1, 0, y4, y5, y3, y5]

 

  p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[y5, y6, y4, y2, y3, 0, y1, 0, 0]  

See Matrices
 

 » SYNC'D 1473/65536 , 0.02247619629

 
115 . Coloring, {2, 4, 6, 9}

R: [4, 9, 4, 8, 7, 8, 1, 1, 2]    B: [2, 4, 5, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [1071, 612, 25, 975, 100, 225, 375, 900, 459] . FixedPtCheck, [1071, 612, 25, 975, 100, 225, 375, 900, 459]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y4 - y3 + 5 y5, y1, 0, y2, 0, 0, y4, y3, y5]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{3, 5}},   net cycles: -1 .    order:   6

See Matrix
 

[y4, y5, y2, y3, y1, 2 y4, 3 y4 - y5 - y2 + y3 + y1, 0, 0 ]

 

  p = s 5 - s 7   p' = s 5 - s 6

 » SYNC'D 2001/131072 , 0.01526641846

 
116 . Coloring, {2, 4, 7, 8}

R: [4, 9, 4, 8, 7, 7, 5, 6, 1]    B: [2, 4, 5, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` - 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-481, -148, -215, -559, -860, -387, -1075, -516, -111] . FixedPtCheck, [481, 148, 215, 559, 860, 387, 1075, 516, 111]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y7, 0, 0, y6, y4, y5, y3, y1, y2]  

See Matrices
 

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, y4, y3, -y5 + 2 y3 + 3 y2, y2, 0, y1, -y4 + 3 y3 + 2 y2 - y1, 0]

 

  p' = - s 2 + s 6   p = - s 2 + s 6

 » SYNC'D 52005/2097152 , 0.02479791641

 
117 . Coloring, {2, 4, 7, 9}

R: [4, 9, 4, 8, 7, 7, 5, 1, 2]    B: [2, 4, 5, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-119, -68, -25, -125, -100, -25, -125, -100, -51] . FixedPtCheck, [119, 68, 25, 125, 100, 25, 125, 100, 51]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y2 + y4, -y1 - y5 - y4 - y6, -y4 - y3, y2, y1, y5, y4, y3, y6]
  p = s - 8s 3 - 12s 4 + 32s 6 + 32s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 8 7 vs 8 4 vs 7 4 vs 8

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}, {2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[3 y3 - y4 - y2, y3 - y1, 0, y4, y3, 0, y3, y2, y1]

 

  p' = s 2 + s 3 - s 5 - s 6   p' = s - s 3 - s 4 + s 6   p = s - s 7

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 7}, {6, 8}},   net cycles: 3 .    order:   4

See Matrix
 

[-y1 + 2 y2 + 2 y4, 2 y2 + 2 y4 - y3, y4, y1, y2, y4, y3, y2, 0]

 

  p = - s + s 5   p' = - s + s 5   p' = - s 2 + s 6   p' = - s 3 + s 7


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, 0, -8 x1, -12 x1, 0, 32 x1, 32 x1]
For A+2Δ :   [-26 y1 - 27 y2, 8 y1 + 9 y2, y1, -2 y1 - 3 y2, 8 y1 + 9 y2, 80 y1 + 81 y2, -y1, y1, y2]
For A-2Δ :   [2 y1, -8 y1 - 6 y2, -y1 - 3 y2, 26 y1 + 24 y2, -8 y1 - 6 y2, 2 y2, y1 + 3 y2, -y1 - 3 y2, -80 y1 - 78 y2]

Range of {ΩΔi}: [-μ1 + μ4, -μ4 - μ2 - μ3 - μ6, -μ4 - μ5, μ1, μ2, μ3, μ4, μ5, μ6]

 
rank of M is 8 , rank of N is 2

M              \ ;    N

$ [ [0, 142, 0, 0, 104, 87, 191, 0, 46] , [142, 0, 52, 94, 0, 0, 0, 92, 0] , [0, 52, 0, 0, 0, 57, 81, 0, 0] , [0, 94, 0, 0, 162, 46, 171, 0, 97] , [104, 0, 0, 162, 0, 0, 0, 114, 0] , [87, 0, 57, 46, 0, 0, 0, 0, 0] , [191, 0, 81, 171, 0, 0, 0, 127, 0] , [0, 92, 0, 0, 114, 0, 127, 0, 47] , [46, 0, 0, 97, 0, 0, 0, 47, 0] ] $     $ [ [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [1, 0, 1, 1, 0, 0, 0, 1, 0] , [0, 1, 0, 0, 1, 1, 1, 0, 1] , [1, 0, 1, 1, 0, 0, 0, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [-1, -1, -1, 1, 1, -1, 0, 1, 1]

ker M, [0, -21054 λ1, 0, 0, 25083 λ1, 9544 λ1, 6800 λ1, 0, -38002 λ1]
Range M, [6800 x1, 6800 x2, 6800 x3, 6800 x5, 6800 x4, 6800 x7, 21054 x2 - 25083 x4 - 9544 x7 + 38002 x8, 6800 x6, 6800 x8]

τ= 41 , r'= 1/2

Ranges

Action of R on ranges, [[14], [13], [13], [11], [7], [14], [17], [5], [13], [11], [16], [16], [15], [8], [4], [2], [1]]
Action of B on ranges, [[7], [6], [8], [1], [1], [11], [13], [12], [15], [2], [10], [16], [4], [4], [9], [3], [3]]
β({1, 2}) = 71/855
β({1, 5}) = 52/855
β({1, 6}) = 29/570
β({1, 7}) = 191/1710
β({1, 9}) = 23/855
β({2, 3}) = 26/855
β({2, 4}) = 47/855
β({2, 8}) = 46/855
β({3, 6}) = 1/30
β({3, 7}) = 9/190
β({4, 5}) = 9/95
β({4, 6}) = 23/855
β({4, 7}) = 1/10
β({4, 9}) = 97/1710
β({5, 8}) = 1/15
β({7, 8}) = 127/1710
β({8, 9}) = 47/1710

ker N, [μ5, μ6, μ7, μ2, μ3, μ4, -μ6 - μ3 - μ4 - μ1, -μ5 - μ7 - μ2, μ1]
Range of N
    [y1, y2, y1, y1, y2, y2, y2, y1, y2]

Partitions
α([{2, 5, 6, 7, 9}, {1, 3, 4, 8}]) = 1/1

b1 = {2, 5, 6, 7, 9} ` , ` b2 = {1, 3, 4, 8}

Action of R and B on the blocks of the partitions: = [1, 2] [2, 1]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {2, 4, 7, 9}
Rank2
R,B [4, 9, 4, 8, 7, 7, 5, 1, 2], [2, 4, 5, 7, 3, 8, 1, 6, 1]
π2 [142, 0, 0, 104, 87, 191, 0, 46, 52, 94, 0, 0, 0, 92, 0, 0, 0, 57, 81, 0, 0, 162, 46, 171, 0, 97, 0, 0, 114, 0, 0, 0, 0, 127, 0, 47]
u2 [1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1] (dim 1)
wpp [4, 5, 4, 4, 5, 5, 5, 4, 5]

 

 
118 . Coloring, {2, 4, 8, 9}

R: [4, 9, 4, 8, 7, 7, 1, 6, 2]    B: [2, 4, 5, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 5 + 3τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-2079, -1188, -125, -1950, -500, -1350, -1875, -1800, -891] . FixedPtCheck, [2079, 1188, 125, 1950, 500, 1350, 1875, 1800, 891]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}, {2, 9}},   net cycles: 2 .   

See Matrix
 

[5 y1 - y2 - y3 - y4 - y5 + 5 y6, y1, 0, y2, 0, y3, y4, y5, y6]

 

  p = - s - s 2 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 + y5 + y6, y1, y2, y3, y4, 0, y5, y6, 0]

 

  p = - s 6 + s 7

 » SYNC'D 407263/33554432 , 0.01213738322

 
119 . Coloring, {2, 5, 6, 7}

R: [4, 9, 4, 7, 3, 8, 5, 1, 1]    B: [2, 4, 5, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 387, 559, 516, 43, 559, 172, 75] . FixedPtCheck, [325, 100, 387, 559, 516, 43, 559, 172, 75]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, 0, y1, y3, y5, 0, y4, y6, 2 y6]

 

  p = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 0 .    order:   6

[y2, y1, 0, y4, y3, y6, y5, y7, 0]  

See Matrices
 

 » SYNC'D 140385/8388608 , 0.01673519611

 
120 . Coloring, {2, 5, 6, 8}

R: [4, 9, 4, 7, 3, 8, 1, 6, 1]    B: [2, 4, 5, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-611, -188, -129, -602, -172, -258, -559, -344, -141] . FixedPtCheck, [611, 188, 129, 602, 172, 258, 559, 344, 141]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[-2 y5 - y1 + 5 y2 - y3 + 5 y4, 0, y5, y1, 0, y2, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[5 y3, 5 y4, 0, -5 y3 - 16 y4 + 33 y1 - 16 y2, 5 y1, 0, -7 y4 + 16 y1 - 7 y2, 5 y2, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 3949/262144 , 0.01506423950

 
121 . Coloring, {2, 5, 6, 9}

R: [4, 9, 4, 7, 3, 8, 1, 1, 2]    B: [2, 4, 5, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [1869, 1068, 375, 1950, 500, 150, 1625, 600, 801] . FixedPtCheck, [1869, 1068, 375, 1950, 500, 150, 1625, 600, 801]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y4, y5, 2 y2, y3, 0, 0, -y4 + 5 y5 - y3 - 3 y2 + 5 y1, y2, y1]

 

  p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 4 + s 5 - s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y1, y1 + y2 + y3 + y5 - y6 - y4, 0, y2, y3, y5, y6, y4, 0]

 

  p = - s 6 + s 7

 » SYNC'D 154171/4194304 , 0.03675723076

 
122 . Coloring, {2, 5, 7, 8}

R: [4, 9, 4, 7, 3, 7, 5, 6, 1]    B: [2, 4, 5, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [1274, 392, 2322, 2795, 3096, 645, 3354, 860, 294] . FixedPtCheck, [1274, 392, 2322, 2795, 3096, 645, 3354, 860, 294]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y5 - y1, -y3 - y5, -y3 - y4, -y2 + y3 + y5, y1, y2, y3, y4, y5]
  p = s 3 - 16s 5 + 8s 6 - 32s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s 3 + 4s 4 + 8s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, 0, y3, y2, y1, y6, y5, 0, y6]

 

  p = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y1, y2, 0, y6, y3, 0, y4, y5, 0]  

See Matrices
 

 » SYNC'D 735/32768 , 0.02243041992

 
123 . Coloring, {2, 5, 7, 9}

R: [4, 9, 4, 7, 3, 7, 5, 1, 2]    B: [2, 4, 5, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` , 54` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-966, -552, -1350, -1875, -1800, -125, -1950, -500, -414] . FixedPtCheck, [966, 552, 1350, 1875, 1800, 125, 1950, 500, 414]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 9}, {3, 4, 5, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y2 - y1 - y5 + 3 y4, y2, y1, y3, 3 y2 - y3 + 2 y4, 0, y5, 0, y4]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y1, y1 + y2 + y4 + y5 - y3 - y6, 0, y2, y4, y5, y3, y6, 0]

 

  p = s 6 - s 7

 » SYNC'D 14193/524288 , 0.02707099915

 
124 . Coloring, {2, 5, 8, 9}

R: [4, 9, 4, 7, 3, 7, 1, 6, 2]    B: [2, 4, 5, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [301, 172, 75, 325, 100, 75, 325, 100, 129] . FixedPtCheck, [301, 172, 75, 325, 100, 75, 325, 100, 129]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[5 y3 - 2 y5 - y2 - y1 + 5 y4, y3, y5, y2, 0, y5, y1, 0, y4]

 

  p = - s 2 - s 3 + s 5 + s 6   p' = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[3 y1 - y4 - 4 y3 + 3 y2, y1, 0, y4, y3, 0, 2 y1 - 3 y3 + 2 y2, y2, 0]

 

  p = s - s 5   p' = s - s 5

 » SYNC'D 595/65536 , 0.009078979492

 
125 . Coloring, {2, 6, 7, 8}

R: [4, 9, 4, 7, 7, 8, 5, 6, 1]    B: [2, 4, 5, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-247, -76, -129, -301, -516, -129, -645, -172, -57] . FixedPtCheck, [247, 76, 129, 301, 516, 129, 645, 172, 57]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-15 y5 - y2 + 4 y3 + 4 y4 + 4 y1, 0, 0, y3, y4, y5, y2, y3 + y4 - 4 y5 + y1, y1]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - y2 + 3 y3, 3 y1 + 2 y3 - y4 - y5, y1, y2, y3, 0, y4, y5, 0]

 

  p = s 2 - s 6   p' = - s 2 + s 6

 » SYNC'D 30183/2097152 , 0.01439237595

 
126 . Coloring, {2, 6, 7, 9}

R: [4, 9, 4, 7, 7, 8, 5, 1, 2]    B: [2, 4, 5, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-259, -148, -125, -325, -500, -25, -625, -100, -111] . FixedPtCheck, [259, 148, 125, 325, 500, 25, 625, 100, 111]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y2 - y4 + 2 y3, y2, 0, y1, 2 y2 - y1 - y5 + 3 y3, 0, y4, y5, y3]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 6, 7, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y6, y1, y2, -y6 + 3 y2 + 2 y3 - y4, y3, y4, y5, -y1 + 2 y2 + 3 y3 - y5, 0]

 

  p = - s + s 7   p' = - s + s 7

 » SYNC'D 162285/8388608 , 0.01934587955

 
127 . Coloring, {2, 6, 8, 9}

R: [4, 9, 4, 7, 7, 8, 1, 6, 2]    B: [2, 4, 5, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` - 5 - 3τ - τ 2 + τ 3 ` )` , -9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 - 3τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-371, -212, -25, -350, -100, -150, -375, -200, -159] . FixedPtCheck, [371, 212, 25, 350, 100, 150, 375, 200, 159]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}, {6, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[4 y3 - y1 + 4 y4 - y2, y3, 0, y1, 0, y4, y2, y3, y4]

 

  p = s + s 2 - s 4 - s 5   p = s - s 3 - s 4 + s 6   p = - s + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y4, 2 y3, 2 y2, -2 y4 + 9 y3 - 11 y1 + 9 y5, 2 y1, 0, 7 y3 - 2 y2 - 9 y1 + 7 y5, 2 y5, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 111573/33554432 , 0.003325134516

 
128 . Coloring, {2, 7, 8, 9}

R: [4, 9, 4, 7, 7, 7, 5, 6, 2]    B: [2, 4, 5, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-238, -136, -150, -325, -600, -75, -750, -100, -102] . FixedPtCheck, [238, 136, 150, 325, 600, 75, 750, 100, 102]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 3 vs 6 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 0 .    order:   2

See Matrix
 

[0, y1 + 3 y2 - 4 y3, 0, 2 y2, y1, y2, 4 y1 + 12 y2 - 15 y3, 0, y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {3, 5}},   net cycles: 2 .    order:   4

See Matrix
 

[3 y1 - y2 + 2 y3, 2 y1 + 3 y3 - y4, y1, y2, y3, 0, 0, y4, 0]

 

  p' = s - s 5   p = - s + s 5

 » SYNC'D 795/32768 , 0.02426147461

 
129 . Coloring, {3, 4, 5, 6}

R: [4, 4, 5, 8, 3, 8, 1, 1, 1]    B: [2, 9, 4, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-750, -200, -114, -741, -152, -171, -266, -684, -50] . FixedPtCheck, [750, 200, 114, 741, 152, 171, 266, 684, 50]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 3 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y3, 0, y4, -y3 + 5 y4 + 5 y1 - y2, y1, 0, 0, y2, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 0 .    order:   2

See Matrix
 

[0, y3 - y2, 0, y1, -3 y1 + y3, 2 y1, y3, 0, y2]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6

 » SYNC'D 121/8192 , 0.01477050781

 
130 . Coloring, {3, 4, 5, 7}

R: [4, 4, 5, 8, 3, 7, 5, 1, 1]    B: [2, 9, 4, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-465, -124, -513, -570, -684, -114, -399, -456, -31] . FixedPtCheck, [465, 124, 513, 570, 684, 114, 399, 456, 31]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[11 y1 - 7 y2 + 11 y3 + 11 y4 - 7 y5, 0, 7 y1, 7 y2, 7 y3, 0, 7 y4, 7 y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-y1 + y2 + 4 y4 - y5, 4 y2 - y3 + y4, 0, y1, 0, y2, y3, y4, y5]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

 » SYNC'D 51985/2097152 , 0.02478837967

 
131 . Coloring, {3, 4, 5, 8}

R: [4, 4, 5, 8, 3, 7, 1, 6, 1]    B: [2, 9, 4, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-705, -188, -285, -741, -380, -513, -665, -684, -47] . FixedPtCheck, [705, 188, 285, 741, 380, 513, 665, 684, 47]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 4 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 6, 7, 8}},   net cycles: 2 .   

See Matrix
 

[5 y1 - y4 + 5 y5 - y6 - y3 - y2, 0, y1, y4, y5, y6, y3, y2, 0]

 

  p = - s - s 2 + s 6 + s 7

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 0 .    order:   4

See Matrix
 

[y4, y2, 0, y3, y1, 0, y2 + y3, y3, -y4 + y1 + y3]

 

  p' = s 3 - s 5   p = - s 3 + s 5   p = - s 3 + s 7

 » SYNC'D 2821/262144 , 0.01076126099

 
132 . Coloring, {3, 4, 5, 9}

R: [4, 4, 5, 8, 3, 7, 1, 1, 2]    B: [2, 9, 4, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-247, -76, -51, -255, -68, -51, -119, -204, -19] . FixedPtCheck, [247, 76, 51, 255, 68, 51, 119, 204, 19]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 8

Omega Rank for R :  cycles: {{1, 4, 8}, {3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[-2 y4 + 5 y1 - y3 + 5 y2 - y5, y4, y1, y3, y2, 0, y4, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[-y1 - 2 y3 + 2 y4 - y2, y1, 0, -y5 - 3 y3 + 2 y4, y5, y3, y4, -2 y3 + y4, y2]

 

  p = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 8   p = s 2 - s 4 - s 5 + s 7

 » SYNC'D 119533/16777216 , 0.007124722004

 
133 . Coloring, {3, 4, 6, 7}

R: [4, 4, 5, 8, 7, 8, 5, 1, 1]    B: [2, 9, 4, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-735, -196, -171, -741, -684, -171, -741, -684, -49] . FixedPtCheck, [735, 196, 171, 741, 684, 171, 741, 684, 49]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2 - 3 y3 - y4 - 2 y5 - y1, -2 y2 + 2 y3 - y6 + y4 + 2 y5, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 3 - 3s 4 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 6 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}, {5, 7}},   net cycles: 2 .    order:   6

See Matrix
 

[y4, 0, 0, y2, y3, 0, y1, -y4 - y2 + 2 y3 + 2 y1, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, y2, y3, y5, 0, y3, y4, 0, y6]

 

  p = - s 5 + s 7

 » SYNC'D 2725/65536 , 0.04158020020

 
134 . Coloring, {3, 4, 6, 8}

R: [4, 4, 5, 8, 7, 8, 1, 6, 1]    B: [2, 9, 4, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -54` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-690, -184, -38, -665, -152, -855, -494, -1140, -46] . FixedPtCheck, [690, 184, 38, 665, 152, 855, 494, 1140, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 - y2 + y3 + y4 - y5, 0, 0, y1, y2, y3, y4, y5, 0]

 

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - 2 y3, 2 y1, 5 y1 - 2 y2 - 2 y4 - 2 y5, 2 y2, 2 y4, 0, 2 y5, 0, 2 y3]

 

  p' = - s 2 + s 6   p = - s 2 + s 6

 » SYNC'D 15171/1048576 , 0.01446819305

 
135 . Coloring, {3, 4, 6, 9}

R: [4, 4, 5, 8, 7, 8, 1, 1, 2]    B: [2, 9, 4, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [1118, 344, 34, 1105, 136, 255, 442, 1020, 86] . FixedPtCheck, [1118, 344, 34, 1105, 136, 255, 442, 1020, 86]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 6 6 vs 8

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y1, y2, 0, y1 + y2 - y3, y3, 0]

 

  p = s 3 - s 4   p' = - s 3 + s 4   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{3, 4, 5, 7}, {1, 2, 9}},   net cycles: 1 .   

See Matrix
 

[y5, y4, y3, y2, -y2 - y1 + y5 + y4 + y6, y1, y5 + y4 - y3 + y6, 0, y6]

 

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7

 » SYNC'D 24189/4194304 , 0.005767107010

 
136 . Coloring, {3, 4, 7, 8}

R: [4, 4, 5, 8, 7, 7, 5, 6, 1]    B: [2, 9, 4, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [1335, 356, 855, 1482, 3420, 1026, 3705, 1368, 89] . FixedPtCheck, [1335, 356, 855, 1482, 3420, 1026, 3705, 1368, 89]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y1, 0, 0, y2, y3, y4, y5, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, y2, 2 y5, y3, 0, 0, y4, y5, y6]

 

  p = - s 5 + s 7

 » SYNC'D 6991/65536 , 0.1066741943

 
137 . Coloring, {3, 4, 7, 9}

R: [4, 4, 5, 8, 7, 7, 5, 1, 2]    B: [2, 9, 4, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-481, -148, -153, -510, -612, -102, -663, -408, -37] . FixedPtCheck, [481, 148, 153, 510, 612, 102, 663, 408, 37]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )`` (` - 1 + τ ` )` 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[7 y4, 7 y3, 0, -7 y4 - 7 y3 + 11 y2 + 11 y1 - 7 y5, 7 y2, 0, 7 y1, 7 y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y1, -y1 - y2 - y3 + 5 y4 - y5 + 5 y6 - y7, y2, y3, 0, y4, y5, y6, y7]

 

  p = s 4 + s 5 - s 7 - s 8

 » SYNC'D 1537181/33554432 , 0.04581156373

 
138 . Coloring, {3, 4, 8, 9}

R: [4, 4, 5, 8, 7, 7, 1, 6, 2]    B: [2, 9, 4, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [13, 4, 1, 13, 4, 9, 13, 12, 1] . FixedPtCheck, [13, 4, 1, 13, 4, 9, 13, 12, 1]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 5 vs 8

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: -1 .    order:   5

See Matrix
 

[y1, y6, 0, y2, y6, y3, y4, y5, 0]

 

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{1, 2, 9}, {3, 4, 5, 7}},   net cycles: 1 .   

See Matrix
 

[y5, y5, y4, y3, y2, 0, 3 y5 - y4 - y3 - y2, y1, y5 - y1]

 

  p = s 2 - s 6   p' = - s 3 + s 7   p' = s 2 - s 6

 » SYNC'D 449775/67108864 , 0.006702169776

 
139 . Coloring, {3, 5, 6, 7}

R: [4, 4, 5, 7, 3, 8, 5, 1, 1]    B: [2, 9, 4, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 18` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )` 3 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [-885, -236, -2565, -1482, -3420, -114, -1995, -456, -59] . FixedPtCheck, [885, 236, 2565, 1482, 3420, 114, 1995, 456, 59]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[y2, 0, y1, y3, y5, 0, y4, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   6

[y1, y2, 0, y3, 0, y4, y5, y6, y7]  

See Matrices
 

 » SYNC'D 50733/2097152 , 0.02419137955

 
140 . Coloring, {3, 5, 6, 8}

R: [4, 4, 5, 7, 3, 8, 1, 6, 1]    B: [2, 9, 4, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-375, -100, -171, -399, -228, -171, -399, -228, -25] . FixedPtCheck, [375, 100, 171, 399, 228, 171, 399, 228, 25]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}, {6, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[4 y4 - y1 - y3 + 4 y2, 0, y4, y1, y2, y4, y3, y2, 0]

 

  p' = s - s 3 - s 4 + s 6   p' = s 2 + s 3 - s 5 - s 6   p = s - s 7

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, y4, 0, y3, y2, 0, y2, y1, -y5 - y4 - y3 + 4 y2 - y1]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

 » SYNC'D 4455/2097152 , 0.002124309540

 
141 . Coloring, {3, 5, 6, 9}

R: [4, 4, 5, 7, 3, 8, 1, 1, 2]    B: [2, 9, 4, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [611, 188, 255, 663, 340, 51, 595, 204, 47] . FixedPtCheck, [611, 188, 255, 663, 340, 51, 595, 204, 47]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[-2 y5 + 5 y1 - y2 + 5 y3 - y4, y5, y1, y2, y3, 0, y4, y5, 0]

 

  p' = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y6 + y5 + y3 + y4 - y1, y6, 0, y5, y3, y4, y2, y5 + y3 + y4 - y2, y1]

 

  p = - s 4 + s 7   p' = - s 4 + s 7

 » SYNC'D 45045/4194304 , 0.01073956490

 
142 . Coloring, {3, 5, 7, 8}

R: [4, 4, 5, 7, 3, 7, 5, 6, 1]    B: [2, 9, 4, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 4 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 4 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-375, -100, -1539, -741, -2052, -171, -1197, -228, -25] . FixedPtCheck, [375, 100, 1539, 741, 2052, 171, 1197, 228, 25]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y4, y3, -2 y4 - y3 - 2 y1 + y5 - y6, y1, y2, -y4 - y3 - y1 - y2 - y6, 2 y4 + y3 + 2 y1 - 2 y5 + y6, y5, y6]
  p = s 2 + 2s 4 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   4

See Matrix
 

[y3, 0, y2, y1, y5, 2 y3, y4, 0, 0]

 

  p = s 4 - s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y3, y4, 0, y5, 0, 0, 2 y5, y1, y2]

 

  p = - s 4 + s 6

 » SYNC'D 13/128 , 0.1015625000

 
143 . Coloring, {3, 5, 7, 9}

R: [4, 4, 5, 7, 3, 7, 5, 1, 2]    B: [2, 9, 4, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , -18` (` 5 + τ ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + τ ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [143, 44, 459, 255, 612, 17, 357, 68, 11] . FixedPtCheck, [143, 44, 459, 255, 612, 17, 357, 68, 11]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 7

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y1, y1, -3 y1 + y2 + y3 - y4, y2, y3, 0, y4, 0, 0]

 

  p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[3 y1, 3 y2, 0, -7 y1 - 7 y2 + 11 y3 + 11 y4 - 7 y5, 0, 3 y3, -14 y1 - 14 y2 + 22 y3 + 22 y4 - 14 y5, 3 y4, 3 y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

 » SYNC'D 4245/262144 , 0.01619338989

 
144 . Coloring, {3, 5, 8, 9}

R: [4, 4, 5, 7, 3, 7, 1, 6, 2]    B: [2, 9, 4, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 3 + τ 2 ` )` , -54` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [598, 184, 306, 663, 408, 153, 714, 204, 46] . FixedPtCheck, [598, 184, 306, 663, 408, 153, 714, 204, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y4, y5, y2, y3, y1, 2 y5, -y4 - 3 y5 + 5 y2 - y3 + 5 y1, 0, 0]

 

  p = s 2 - s 4 - s 5 + s 7   p' = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1, -5 y1 - 5 y6 + 13 y5 + 13 y4 - 5 y3 - 5 y2, 0, 5 y6, 5 y5, 0, 5 y4, 5 y3, 5 y2]

 

  p = - s 3 - s 4 + s 6 + s 7

 » SYNC'D 86953/2097152 , 0.04146242142

 
145 . Coloring, {3, 6, 7, 8}

R: [4, 4, 5, 7, 7, 8, 5, 6, 1]    B: [2, 9, 4, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` 2 , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [705, 188, 513, 798, 2052, 342, 2223, 456, 47] . FixedPtCheck, [705, 188, 513, 798, 2052, 342, 2223, 456, 47]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y2 - y3 + 4 y4, 0, 0, -y1 + 4 y2 + y4, y1, y2, y3, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y3, y4, 2 y1, y5, 0, 0, y1, y2, y6]

 

  p = - s 5 + s 7

 » SYNC'D 41/1024 , 0.04003906250

 
146 . Coloring, {3, 6, 7, 9}

R: [4, 4, 5, 7, 7, 8, 5, 1, 2]    B: [2, 9, 4, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-1157, -356, -765, -1326, -3060, -102, -3315, -408, -89] . FixedPtCheck, [1157, 356, 765, 1326, 3060, 102, 3315, 408, 89]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 4 ` (` τ ` )` 2 ` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y5, y4, 0, y3, y2, 0, y1, y4, 0]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y8, y7, y6, y5, 0, y4, y3, y2, y1]  

See Matrices
 

 » SYNC'D 2262579/67108864 , 0.03371505439

 
147 . Coloring, {3, 6, 8, 9}

R: [4, 4, 5, 7, 7, 8, 1, 6, 2]    B: [2, 9, 4, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , -18` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + 4τ + 6τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [1781, 548, 153, 1785, 612, 765, 1989, 1020, 137] . FixedPtCheck, [1781, 548, 153, 1785, 612, 765, 1989, 1020, 137]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[-y1 - 2 y2 + 5 y3 - y4 + 5 y5, y2, 0, y1, y2, y3, y4, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y5, y4, y3, y2, y1, 0, y8, y7, y6]  

See Matrices
 

 » SYNC'D 947699/33554432 , 0.02824363112

 
148 . Coloring, {3, 7, 8, 9}

R: [4, 4, 5, 7, 7, 7, 5, 6, 2]    B: [2, 9, 4, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-559, -172, -459, -663, -1836, -153, -1989, -204, -43] . FixedPtCheck, [559, 172, 459, 663, 1836, 153, 1989, 204, 43]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 6 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y1, 0, y2, y3, 2 y1, y4, 0, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y4, y3, y2, y1, 0, 0, 0, y5, y6]  

See Matrices
 

 » SYNC'D 537/4096 , 0.1311035156

 
149 . Coloring, {4, 5, 6, 7}

R: [4, 4, 4, 8, 3, 8, 5, 1, 1]    B: [2, 9, 5, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -27` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-3090, -824, -1026, -3705, -1368, -855, -1482, -3420, -206] . FixedPtCheck, [3090, 824, 1026, 3705, 1368, 855, 1482, 3420, 206]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 5 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: 0 .    order:   3

[y4, 0, y3, y2, y1, 0, 0, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y3, y4, 0, 0, y5, 2 y5, y1, 0, y2]

 

  p = - s 4 + s 6

 » SYNC'D 19/256 , 0.07421875000

 
150 . Coloring, {4, 5, 6, 8}

R: [4, 4, 4, 8, 3, 8, 1, 6, 1]    B: [2, 9, 5, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-375, -100, -57, -399, -76, -513, -247, -684, -25] . FixedPtCheck, [375, 100, 57, 399, 76, 513, 247, 684, 25]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y2 + y3 + y4 - y1, 2 y2 - 2 y3 - y6 - y4, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 3 - s 4 + 4s 5 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 3 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[2 y1, 0, y1, y2, 0, y3, 0, y4, 0]

 

  p = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   2

See Matrix
 

[2 y1, 2 y1 + 2 y2, 0, 0, 2 y3, 0, 5 y1 + 5 y2 - 2 y3, 0, 2 y2]

 

  p' = s 2 - s 4   p = s 2 - s 4

 » SYNC'D 1/64 , 0.01562500000

 
151 . Coloring, {4, 5, 6, 9}

R: [4, 4, 4, 8, 3, 8, 1, 1, 2]    B: [2, 9, 5, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-637, -196, -51, -663, -68, -153, -221, -612, -49] . FixedPtCheck, [637, 196, 51, 663, 68, 153, 221, 612, 49]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 5 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y1, 2 y1, y4, 0, 0, 0, y3, 0]

 

  p = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y2 + y1 - y4, y2, 0, 0, -y3 + y1, y3, y1, 0, y4]

 

  p = - s 2 + s 5   p' = - s 2 + s 5

 » SYNC'D 4005/131072 , 0.03055572510

 
152 . Coloring, {4, 5, 7, 8}

R: [4, 4, 4, 8, 3, 7, 5, 6, 1]    B: [2, 9, 5, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-375, -100, -513, -741, -684, -513, -741, -684, -25] . FixedPtCheck, [375, 100, 513, 741, 684, 513, 741, 684, 25]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 8}},   net cycles: 0 .    order:   6

[y4, 0, y5, y1, y2, y3, y6, y7, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y1, 0, 0, y5, 0, y3, y5, y4]

 

  p = s 4 - s 6

 » SYNC'D 5415/65536 , 0.08262634277

 
153 . Coloring, {4, 5, 7, 9}

R: [4, 4, 4, 8, 3, 7, 5, 1, 2]    B: [2, 9, 5, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 18` (` - 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [-949, -292, -459, -1275, -612, -255, -663, -1020, -73] . FixedPtCheck, [949, 292, 459, 1275, 612, 255, 663, 1020, 73]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   6

See Matrix
 

[y4, y6, y3, y2, y1, 0, y6, y5, 0]

 

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 - y2 + 5 y3 - y4 + 5 y5 - y6, y1, 0, 0, y2, y3, y4, y5, y6]

 

  p = - s 3 - s 4 + s 6 + s 7

 » SYNC'D 125277/2097152 , 0.05973672867

 
154 . Coloring, {4, 5, 8, 9}

R: [4, 4, 4, 8, 3, 7, 1, 6, 2]    B: [2, 9, 5, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [1157, 356, 255, 1326, 340, 918, 1105, 1224, 89] . FixedPtCheck, [1157, 356, 255, 1326, 340, 918, 1105, 1224, 89]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 3 vs 6

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}},   net cycles: -1 .    order:   5

See Matrix
 

[y3, y4, 2 y4, y1, 0, y2, y5, y6, 0]

 

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[y2 + y1, y2 + y1, 0, 0, y3, 0, 3 y2 + 3 y1 - y3, y2, y1]

 

  p = - s 2 + s 6   p' = - s 2 + s 4   p = - s 2 + s 4

 » SYNC'D 58695/2097152 , 0.02798795700

 
155 . Coloring, {4, 6, 7, 8}

R: [4, 4, 4, 8, 7, 8, 5, 6, 1]    B: [2, 9, 5, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 + τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-330, -88, -114, -399, -456, -513, -570, -684, -22] . FixedPtCheck, [330, 88, 114, 399, 456, 513, 570, 684, 22]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, 0, y3, -5 y1 + 14 y2 - 5 y4, -14 y1 - y3 + 39 y2 - 14 y4, y2, y4, 0]

 

  p = s 3 - s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 + 3 y2 - y3, 3 y1 + 2 y2 - y4, y1, 0, y2, 0, y4, 0, y3]

 

  p = s 3 - s 5   p' = s 3 - s 5

 » SYNC'D 233/32768 , 0.007110595703

 
156 . Coloring, {4, 6, 7, 9}

R: [4, 4, 4, 8, 7, 8, 5, 1, 2]    B: [2, 9, 5, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 3 + τ 2 ` )` , -54` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [598, 184, 102, 663, 408, 153, 510, 612, 46] . FixedPtCheck, [598, 184, 102, 663, 408, 153, 510, 612, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-5 y1 - 5 y2 + 13 y3 + 13 y4 - 5 y5, 5 y1, 0, 5 y2, 5 y3, 0, 5 y4, 5 y5, 0]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 5 y2 + 5 y3 - y4 - y5 - y6, y1, y2, 0, y3, y4, y5, 0, y6]

 

  p = - s 3 - s 4 + s 6 + s 7

 » SYNC'D 18465/524288 , 0.03521919250

 
157 . Coloring, {4, 6, 8, 9}

R: [4, 4, 4, 8, 7, 8, 1, 6, 2]    B: [2, 9, 5, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 ` (` - 5 + τ ` )``]`

For τ=1/2, [351, 108, 17, 357, 68, 459, 255, 612, 27] . FixedPtCheck, [351, 108, 17, 357, 68, 459, 255, 612, 27]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, 0, y3, 0, y4, 2 y2, y5, 0]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[4 y1, 4 y2, 4 y3, 0, 5 y1 + 5 y2 - 4 y3 - 4 y5 + 5 y4, 0, 4 y5, 0, 4 y4]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 105/4096 , 0.02563476562

 
158 . Coloring, {4, 7, 8, 9}

R: [4, 4, 4, 8, 7, 7, 5, 6, 2]    B: [2, 9, 5, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-481, -148, -255, -663, -1020, -459, -1275, -612, -37] . FixedPtCheck, [481, 148, 255, 663, 1020, 459, 1275, 612, 37]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 6 5 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[0, y1, 0, y2, y3, y4, y5, y6, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[y2, -y2 + 5 y1 + 5 y3 - 4 y4 - y5, y1, 0, y3, 0, 3 y4, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

 » SYNC'D 34245/524288 , 0.06531715393

 
159 . Coloring, {5, 6, 7, 8}

R: [4, 4, 4, 7, 3, 8, 5, 6, 1]    B: [2, 9, 5, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 4 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-885, -236, -1539, -1995, -2052, -855, -2223, -1140, -59] . FixedPtCheck, [885, 236, 1539, 1995, 2052, 855, 2223, 1140, 59]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-y1 + y5 + 4 y4 - y3, 0, y1, -y2 + 4 y5 + y4, y2, y5, y3, y4, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, 0, 0, y3, 0, y5, 3 y3, y4]

 

  p = - s 4 + s 6

 » SYNC'D 7689/131072 , 0.05866241455

 
160 . Coloring, {5, 6, 7, 9}

R: [4, 4, 4, 7, 3, 8, 5, 1, 2]    B: [2, 9, 5, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 459, 663, 612, 51, 663, 204, 25] . FixedPtCheck, [325, 100, 459, 663, 612, 51, 663, 204, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y6, y2, y3, y4, 0, y5, y6, 0]

 

  p = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y4, y5, 3 y3, y6]

 

  p = s 4 - s 7

 » SYNC'D 64197/4194304 , 0.01530575752

 
161 . Coloring, {5, 6, 8, 9}

R: [4, 4, 4, 7, 3, 8, 1, 6, 2]    B: [2, 9, 5, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [611, 188, 153, 714, 204, 306, 663, 408, 47] . FixedPtCheck, [611, 188, 153, 714, 204, 306, 663, 408, 47]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[-3 y1 - y5 + 5 y4 - y2 + 5 y3, y1, 2 y1, y5, 0, y4, y2, y3, 0]

 

  p' = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[-7 y2 + 11 y1 + 11 y4 - 7 y5 - 7 y3, 7 y2, 0, 0, 7 y1, 0, 7 y4, 7 y5, 7 y3]

 

  p = - s 2 - s 3 + s 5 + s 6

 » SYNC'D 85825/2097152 , 0.04092454910

 
162 . Coloring, {5, 7, 8, 9}

R: [4, 4, 4, 7, 3, 7, 5, 6, 2]    B: [2, 9, 5, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 4 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [1274, 392, 2754, 3315, 3672, 765, 3978, 1020, 98] . FixedPtCheck, [1274, 392, 2754, 3315, 3672, 765, 3978, 1020, 98]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y2, y1, y3, y4, 2 y2, y5, 0, 0]

 

  p = s 2 - s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y3, y2, 0, 0, y1, 0, y5, 4 y1, y4]

 

  p = s 3 - s 6

 » SYNC'D 4005/65536 , 0.06111145020

 
163 . Coloring, {6, 7, 8, 9}

R: [4, 4, 4, 7, 7, 8, 5, 6, 2]    B: [2, 9, 5, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )``]`

For τ=1/2, [-247, -76, -153, -357, -612, -153, -765, -204, -19] . FixedPtCheck, [247, 76, 153, 357, 612, 153, 765, 204, 19]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y3 - y1 + 4 y4, 0, -y2 + 4 y3 + y4, y2, y3, y1, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[y1, -y1 + 5 y4 + 5 y3 - 4 y2 - y5, y4, 0, y3, 0, y2, 3 y2, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

 » SYNC'D 1265/32768 , 0.03860473633

 
164 . Coloring, {2, 3, 4, 5, 6}

R: [4, 9, 5, 8, 3, 8, 1, 1, 1]    B: [2, 4, 4, 7, 7, 7, 5, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [1950, 600, 258, 1677, 344, 387, 602, 1548, 450] . FixedPtCheck, [1950, 600, 258, 1677, 344, 387, 602, 1548, 450]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 5

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y4 + 5 y3 - y2 - y5, 0, y1, y4, y3, 0, 0, y2, y5]

 

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[0, 2 y3, 0, y1, y2, y3, y4, 0, 0]

 

  p = - s 3 + s 5

 » SYNC'D 269/4096 , 0.06567382812

 
165 . Coloring, {2, 3, 4, 5, 7}

R: [4, 9, 5, 8, 3, 7, 5, 1, 1]    B: [2, 4, 4, 7, 7, 8, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , 18` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-403, -124, -387, -430, -516, -86, -301, -344, -93] . FixedPtCheck, [403, 124, 387, 430, 516, 86, 301, 344, 93]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[3 y5, 0, 3 y3, 3 y4, 3 y2, 0, -7 y5 + 11 y3 - 7 y4 + 11 y2 - 7 y1, 3 y1, -14 y5 + 22 y3 - 14 y4 + 22 y2 - 14 y1]

 

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[y3, y4, 0, -y3 + 4 y4 - 15 y2 + 4 y1, 0, y2, y1, y4 - 4 y2 + y1, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 8581/524288 , 0.01636695862

 
166 . Coloring, {2, 3, 4, 5, 8}

R: [4, 9, 5, 8, 3, 7, 1, 6, 1]    B: [2, 4, 4, 7, 7, 8, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [1833, 564, 645, 1677, 860, 1161, 1505, 1548, 423] . FixedPtCheck, [1833, 564, 645, 1677, 860, 1161, 1505, 1548, 423]

det(A + τ Δ) =   1` (` 1 + τ ` )` 4 ` (` τ ` )` 2 ` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 8 6 vs 6

Omega Rank for R :  cycles: {{1, 4, 6, 7, 8}, {3, 5}},   net cycles: 1 .   

See Matrix
 

[5 y2 - y1 + 5 y5 - y6 - y7 - y4 - y3, 0, y2, y1, y5, y6, y7, y4, y3]

 

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y1, y2, 0, y3, y4, 0, y5, y6, 0]  

See Matrices
 

 » SYNC'D 223785/8388608 , 0.02667725086

 
167 . Coloring, {2, 3, 4, 5, 9}

R: [4, 9, 5, 8, 3, 7, 1, 1, 2]    B: [2, 4, 4, 7, 7, 8, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )``]`

For τ=1/2, [-399, -228, -75, -375, -100, -75, -175, -300, -171] . FixedPtCheck, [399, 228, 75, 375, 100, 75, 175, 300, 171]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[4 y2 + 4 y5 - y1 - y3 - y4, y2, y5, y1, y2, 0, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-y5 - y4 + 2 y3 + 3 y1, 3 y3 - y2 + 2 y1, 0, y5, y4, y3, y2, y1, 0]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

 » SYNC'D 53229/16777216 , 0.003172695637

 
168 . Coloring, {2, 3, 4, 6, 7}

R: [4, 9, 5, 8, 7, 8, 5, 1, 1]    B: [2, 4, 4, 7, 3, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [637, 196, 129, 559, 516, 129, 559, 516, 147] . FixedPtCheck, [637, 196, 129, 559, 516, 129, 559, 516, 147]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y4 - 2 y5 + y2 - 3 y3 - 2 y6 - y1, y4 + 2 y5 - 2 y2 + 2 y3 + y6, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 2 + 2s 4 - 8s 5 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 2 y2 + 2 y3 - y4 - y5, 0, 0, y1, y2, 0, y3, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y5, y3, y4, y2, 0, y4, y1, 0, 0]

 

  p = - s 2 + s 6

 » SYNC'D 165/4096 , 0.04028320312

 
169 . Coloring, {2, 3, 4, 6, 8}

R: [4, 9, 5, 8, 7, 8, 1, 6, 1]    B: [2, 4, 4, 7, 3, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` - 1 + τ ` )`` (` 5 + 3τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 54` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1794, 552, 86, 1505, 344, 1935, 1118, 2580, 414] . FixedPtCheck, [1794, 552, 86, 1505, 344, 1935, 1118, 2580, 414]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 6

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   6

See Matrix
 

[y1 + y2 + y3 + y5 - y4, 0, 0, y1, y2, y3, y5, y4, 2 y2]

 

  p = - s 5 + s 7   p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

[y3, y2, y1, y5, y6, 0, y4, 0, 0]  

See Matrices
 

 » SYNC'D 1059/32768 , 0.03231811523

 
170 . Coloring, {2, 3, 4, 6, 9}

R: [4, 9, 5, 8, 7, 8, 1, 1, 2]    B: [2, 4, 4, 7, 3, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1806, 1032, 50, 1625, 200, 375, 650, 1500, 774] . FixedPtCheck, [1806, 1032, 50, 1625, 200, 375, 650, 1500, 774]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y3 - y4 - y5 + 5 y6, y1, 0, y2, y3, 0, y4, y5, y6]

 

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 3 y1 - y2 + y3 + y4 - y5, y2, y3, y4, 2 y1, y5, 0, 0 ]

 

  p = - s 3 + s 7   p = - s 3 + s 4 - s 5 + s 6

 » SYNC'D 27669/2097152 , 0.01319360733

 
171 . Coloring, {2, 3, 4, 7, 8}

R: [4, 9, 5, 8, 7, 7, 5, 6, 1]    B: [2, 4, 4, 7, 3, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )``]`

For τ=1/2, [1157, 356, 645, 1118, 2580, 774, 2795, 1032, 267] . FixedPtCheck, [1157, 356, 645, 1118, 2580, 774, 2795, 1032, 267]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[y5, 0, 0, y4, y3, y2, y1, y6, y7]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, y3, 2 y5, y2, 0, 0, y1, y5, 0]

 

  p = s 2 - s 6

 » SYNC'D 6933/131072 , 0.05289459229

 
172 . Coloring, {2, 3, 4, 7, 9}

R: [4, 9, 5, 8, 7, 7, 5, 1, 2]    B: [2, 4, 4, 7, 3, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )` , 18` (` - 5 + τ - τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 5 + τ - τ 2 + τ 3 ` )``]`

For τ=1/2, [-259, -148, -75, -250, -300, -50, -325, -200, -111] . FixedPtCheck, [259, 148, 75, 250, 300, 50, 325, 200, 111]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}, {2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[-2 y1 + 8 y2 - 2 y3 - 8 y4, 3 y2 - 5 y4, 0, 2 y1, 2 y2, 0, 5 y2 - 7 y4, 2 y3, 2 y4]

 

  p = - s + s 7   p = - s - s 2 + s 4 + s 5   p' = s + s 2 - s 4 - s 5

Omega Rank for B :  cycles: {{6, 8}, {1, 2, 4, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[-y1 + 2 y5 + 3 y4, y2, -y2 + 3 y5 - y3 + 2 y4, y1, 0, y5, y3, y4, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 55359/16777216 , 0.003299653530

 
173 . Coloring, {2, 3, 4, 8, 9}

R: [4, 9, 5, 8, 7, 7, 1, 6, 2]    B: [2, 4, 4, 7, 3, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [357, 204, 25, 325, 100, 225, 325, 300, 153] . FixedPtCheck, [357, 204, 25, 325, 100, 225, 325, 300, 153]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 3 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 6, 7, 8}},   net cycles: 1 .   

See Matrix
 

[y3, y3 - y2, 0, y3, y3 - y1, y1, y3, y3, y2]

 

  p' = - s 2 + s 6   p = - s 2 + s 6   p = - s 2 + s 8   p' = - s 2 + s 4   p = - s 2 + s 4

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 - y3 - y4 + y5 + y6, y1, y2, y3, y4, 0, y5, y6, 0]

 

  p = - s 4 + s 5 - s 6 + s 7

 » SYNC'D 598125/33554432 , 0.01782551408

 
174 . Coloring, {2, 3, 5, 6, 7}

R: [4, 9, 5, 7, 3, 8, 5, 1, 1]    B: [2, 4, 4, 8, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , 18` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-767, -236, -1935, -1118, -2580, -86, -1505, -344, -177] . FixedPtCheck, [767, 236, 1935, 1118, 2580, 86, 1505, 344, 177]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 7 4 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, 0, y3, y2, y5, 0, y4, y6, 2 y6]

 

  p = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y3 + y1 - y2, -y4 + y3 + y1, 0, y3, 0, y4, y1, y2, 0]

 

  p = s - s 3 + s 4 - s 6   p' = s - s 2 + s 4 - s 5

 » SYNC'D 4725/262144 , 0.01802444458

 
175 . Coloring, {2, 3, 5, 6, 8}

R: [4, 9, 5, 7, 3, 8, 1, 6, 1]    B: [2, 4, 4, 8, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 129, 301, 172, 129, 301, 172, 75] . FixedPtCheck, [325, 100, 129, 301, 172, 129, 301, 172, 75]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )`` (` 1 + τ ` )` 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 7}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y4, 0, y3, y2, y5, y3, -y4 + 4 y3 - y2 + 4 y5 - y1, y5, y1]

 

  p = - s 2 + s 8   p = - s 2 - s 3 + s 5 + s 6   p' = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, -y1 - y2 + 4 y3 - y4, 0, y2, y3, 0, y3, y4, 0]

 

  p' = - s + s 5   p = - s + s 5

 » SYNC'D 15975/8388608 , 0.001904368401

 
176 . Coloring, {2, 3, 5, 6, 9}

R: [4, 9, 5, 7, 3, 8, 1, 1, 2]    B: [2, 4, 4, 8, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [987, 564, 375, 975, 500, 75, 875, 300, 423] . FixedPtCheck, [987, 564, 375, 975, 500, 75, 875, 300, 423]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 7}, {2, 9}},   net cycles: 2 .    order:   6

See Matrix
 

[y1, y2, y3, -y1 + 4 y2 - y5 - y4 + 4 y3, y2, 0, y5, y4, y3]

 

  p = - s 2 + s 8   p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 4 + s 5 - s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 - y2 - y3 - y4 + y5 + y6, y1, 0, y2, y3, y4, y5, y6, 0]

 

  p = s 6 - s 7

 » SYNC'D 114885/16777216 , 0.006847679615

 
177 . Coloring, {2, 3, 5, 7, 8}

R: [4, 9, 5, 7, 3, 7, 5, 6, 1]    B: [2, 4, 4, 8, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [325, 100, 1161, 559, 1548, 129, 903, 172, 75] . FixedPtCheck, [325, 100, 1161, 559, 1548, 129, 903, 172, 75]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2 + y3 - 2 y6 + 2 y5 + y4 - y1, -2 y2 - 2 y3 + y6 - 2 y5 - y4, -y5 - y4, y1, y2, y3, y4, y5, y6]
  p = s 3 + s 4 + 4s 5 + 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 6 vs 7 5 vs 5

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   6

See Matrix
 

[y3, 0, y1, y2, y4, y6, y5, 0, y6]

 

  p = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y3, y4, 0, y5, 0, 0, y2, y1, 0]  

See Matrices
 

 » SYNC'D 2441/65536 , 0.03724670410

 
178 . Coloring, {2, 3, 5, 7, 9}

R: [4, 9, 5, 7, 3, 7, 5, 1, 2]    B: [2, 4, 4, 8, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , 18` (` 5 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [231, 132, 675, 375, 900, 25, 525, 100, 99] . FixedPtCheck, [231, 132, 675, 375, 900, 25, 525, 100, 99]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[2 y1 - y3 - y2 + 3 y5, y1, y3, y4, 3 y1 - y4 + 2 y5, 0, y2, 0, y5]

 

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{6, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y1, y1 + y5 + y4 - y3 - y2, 0, y5, 0, y4, y3, y2, 0]

 

  p = - s 5 + s 6

 » SYNC'D 8917/524288 , 0.01700782776

 
179 . Coloring, {2, 3, 5, 8, 9}

R: [4, 9, 5, 7, 3, 7, 1, 6, 2]    B: [2, 4, 4, 8, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ ` )`` (` 5 + 3τ 2 ` )` , 54` (` 5 + 3τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 1 + τ ` )`` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [322, 184, 150, 325, 200, 75, 350, 100, 138] . FixedPtCheck, [322, 184, 150, 325, 200, 75, 350, 100, 138]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 8 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {1, 4, 7}},   net cycles: 2 .    order:   6

See Matrix
 

[4 y2 + 4 y5 - y1 - y3 - y4, y2, y5, y1, y2, y3, y4, 0, y5]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[9 y1 - 4 y2 - 13 y3 + 9 y4, 4 y1, 0, 4 y2, 4 y3, 0, 5 y1 - 9 y3 + 5 y4, 4 y4, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 9855/4194304 , 0.002349615097

 
180 . Coloring, {2, 3, 6, 7, 8}

R: [4, 9, 5, 7, 7, 8, 5, 6, 1]    B: [2, 4, 4, 8, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [611, 188, 387, 602, 1548, 258, 1677, 344, 141] . FixedPtCheck, [611, 188, 387, 602, 1548, 258, 1677, 344, 141]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y4 - y1 + 4 y2, 0, 0, y5, y3, y4, y1, y2, -y5 - y3 + 4 y4 + y2]

 

  p = s 4 - s 6   p' = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y1, 2 y5, y3, 0, 0, y5, y4, 0]

 

  p = s 2 - s 6

 » SYNC'D 3367/131072 , 0.02568817139

 
181 . Coloring, {2, 3, 6, 7, 9}

R: [4, 9, 5, 7, 7, 8, 5, 1, 2]    B: [2, 4, 4, 8, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , -9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [-623, -356, -375, -650, -1500, -50, -1625, -200, -267] . FixedPtCheck, [623, 356, 375, 650, 1500, 50, 1625, 200, 267]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y3 + 2 y5, y1, 0, 2 y1 - y2 - y4 + 3 y5, y2, 0, y3, y4, y5]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y2 + y3 - y4 - y1 + y5 + y6, y2, y3, y4, 0, y1, y5, y6, 0]

 

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

 » SYNC'D 1971851/67108864 , 0.02938286960

 
182 . Coloring, {2, 3, 6, 8, 9}

R: [4, 9, 5, 7, 7, 8, 1, 6, 2]    B: [2, 4, 4, 8, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 18` (` 5 + 4τ + 6τ 2 + τ 4 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [959, 548, 75, 875, 300, 375, 975, 500, 411] . FixedPtCheck, [959, 548, 75, 875, 300, 375, 975, 500, 411]

det(A + τ Δ) =   1` (` - 1 + τ ` )`` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y5, y4, 0, y1, y2, y3, -y5 + 4 y4 - y1 - y2 + 4 y3, y4, y3]

 

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 - y3 - y4 + y5 + y6, y1, y2, y3, y4, 0, y5, y6, 0]

 

  p = - s 4 + s 5 - s 6 + s 7

 » SYNC'D 599877/134217728 , 0.004469431937

 
183 . Coloring, {2, 3, 7, 8, 9}

R: [4, 9, 5, 7, 7, 7, 5, 6, 2]    B: [2, 4, 4, 8, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-301, -172, -225, -325, -900, -75, -975, -100, -129] . FixedPtCheck, [301, 172, 225, 325, 900, 75, 975, 100, 129]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 6 4 vs 5

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 0 .    order:   2

See Matrix
 

[0, 2 y3, 0, 3 y2, -30 y3 - 5 y2 + 8 y1, 2 y2, 2 y1, 0, -8 y3 + 2 y1]

 

  p' = - s 2 + s 4   p = - s 2 + s 6   p = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 - y3 + y4, y1, y2, y3, 0, 0, 0, y4, 0]

 

  p = s 2 - s 3 + s 4 - s 5

 » SYNC'D 39/1024 , 0.03808593750

 
184 . Coloring, {2, 4, 5, 6, 7}

R: [4, 9, 4, 8, 3, 8, 5, 1, 1]    B: [2, 4, 5, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` 3 + τ 2 ` )` , 54` (` - 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 27` (` - 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-2678, -824, -774, -2795, -1032, -645, -1118, -2580, -618] . FixedPtCheck, [2678, 824, 774, 2795, 1032, 645, 1118, 2580, 618]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[2 y3, 0, 2 y1, 2 y2, 3 y5, 0, 0, 2 y4, 2 y5]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, 0, y4, y3, 2 y3, y5, 0, 0]

 

  p = - s 2 + s 6

 » SYNC'D 2685/65536 , 0.04096984863

 
185 . Coloring, {2, 4, 5, 6, 8}

R: [4, 9, 4, 8, 3, 8, 1, 6, 1]    B: [2, 4, 5, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [975, 300, 129, 903, 172, 1161, 559, 1548, 225] . FixedPtCheck, [975, 300, 129, 903, 172, 1161, 559, 1548, 225]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-3 y1 + y6 + y4 - 2 y3 - y2, 2 y1 - 2 y6 - y4 + y3, -y4 - y5, y2, y1, y6, y4, y5, y3]
  p = s 2 + 6s 4 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, y5, y2, 0, y3, 0, y4, y5]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y4, y3, 0, y5, y1, 0, y2, 0, 0]  

See Matrices
 

 » SYNC'D 391/8192 , 0.04772949219

 
186 . Coloring, {2, 4, 5, 6, 9}

R: [4, 9, 4, 8, 3, 8, 1, 1, 2]    B: [2, 4, 5, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-1029, -588, -75, -975, -100, -225, -325, -900, -441] . FixedPtCheck, [1029, 588, 75, 975, 100, 225, 325, 900, 441]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y3 - y4 + 5 y5, y1, y2, y3, 0, 0, 0, y4, y5]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y3, 0, y1, y4, 2 y2, 3 y2 - y3 + y1 + y4, 0, 0]

 

  p = - s 4 + s 5   p = - s 4 + s 6

 » SYNC'D 4893/131072 , 0.03733062744

 
187 . Coloring, {2, 4, 5, 7, 8}

R: [4, 9, 4, 8, 3, 7, 5, 6, 1]    B: [2, 4, 5, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 387, 559, 516, 387, 559, 516, 75] . FixedPtCheck, [325, 100, 387, 559, 516, 387, 559, 516, 75]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )`` (` 1 + τ ` )` 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 8 vs 8 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 8}},   net cycles: 0 .    order:   6

[y1, 0, y7, y5, y6, y3, y2, y4, y8]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y3, 0, y2, y5, 0, y4, y5, 0]

 

  p = - s 2 + s 6

 » SYNC'D 455085/16777216 , 0.02712517977

 
188 . Coloring, {2, 4, 5, 7, 9}

R: [4, 9, 4, 8, 3, 7, 5, 1, 2]    B: [2, 4, 5, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 - 4τ + 6τ 2 + τ 4 ` )`` (` 3 + τ ` )` , -18` (` 5 - 4τ + 6τ 2 + τ 4 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 - 4τ + 6τ 2 + τ 4 ` )``]`

For τ=1/2, [-511, -292, -225, -625, -300, -125, -325, -500, -219] . FixedPtCheck, [511, 292, 225, 625, 300, 125, 325, 500, 219]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y4 - y1 - y2 - y3 - y5 - y6 + 5 y7, y4, y1, y2, y3, 0, y5, y6, y7]

 

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[-y1 - y2 + 2 y3 + 3 y5, 3 y3 - y4 + 2 y5, 0, y1, y2, y3, y4, y5, 0]

 

  p' = - s 2 + s 6   p = - s 2 + s 6

 » SYNC'D 184063/16777216 , 0.01097100973

 
189 . Coloring, {2, 4, 5, 8, 9}

R: [4, 9, 4, 8, 3, 7, 1, 6, 2]    B: [2, 4, 5, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [1869, 1068, 375, 1950, 500, 1350, 1625, 1800, 801] . FixedPtCheck, [1869, 1068, 375, 1950, 500, 1350, 1625, 1800, 801]

det(A + τ Δ) =   1` (` 1 + τ ` )` 4 ` (` τ ` )` 2 ` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 8 5 vs 6

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 6, 7, 8}},   net cycles: 1 .   

See Matrix
 

[5 y5 - y6 - y7 - y1 - y2 - y3 + 5 y4, y5, y6, y7, 0, y1, y2, y3, y4]

 

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 - y3 - y2 + y4 + y5, y1, 0, y3, y2, 0, y4, y5, 0]

 

  p = - s 5 + s 6

 » SYNC'D 45069/1048576 , 0.04298114777

 
190 . Coloring, {2, 4, 6, 7, 8}

R: [4, 9, 4, 8, 7, 8, 5, 6, 1]    B: [2, 4, 5, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )``]`

For τ=1/2, [286, 88, 86, 301, 344, 387, 430, 516, 66] . FixedPtCheck, [286, 88, 86, 301, 344, 387, 430, 516, 66]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, 0, -14 y1 - y2 + 39 y3 - 14 y4 - y5, -5 y1 + 14 y3 - 5 y4, y2, y3, y4, y5]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {3, 5}},   net cycles: 2 .    order:   4

See Matrix
 

[y3, 3 y2 + 2 y1 - y4, y2, -y3 + 2 y2 + 3 y1, y1, 0, y4, 0, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 2267/262144 , 0.008647918701

 
191 . Coloring, {2, 4, 6, 7, 9}

R: [4, 9, 4, 8, 7, 8, 5, 1, 2]    B: [2, 4, 5, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ 2 ` )`` (` 3 + τ ` )` , 54` (` 5 + 3τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 5 + 3τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [322, 184, 50, 325, 200, 75, 250, 300, 138] . FixedPtCheck, [322, 184, 50, 325, 200, 75, 250, 300, 138]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 4 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 9}, {1, 4, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[-y2 + 10 y1 - y3 - 10 y4, 3 y1 - 4 y4, 0, y2, y1, 0, 4 y1 - 5 y4, y3, y4]

 

  p = - s - s 2 + s 4 + s 5   p = s - s 3 - s 4 + s 6   p = - s + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y2 + 2 y3 - y4, 2 y1 + 3 y3 - y5, y1, y2, y3, y4, y5, 0, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

 » SYNC'D 877/262144 , 0.003345489502

 
192 . Coloring, {2, 4, 6, 8, 9}

R: [4, 9, 4, 8, 7, 8, 1, 6, 2]    B: [2, 4, 5, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` - 5 + τ ` )``]`

For τ=1/2, [567, 324, 25, 525, 100, 675, 375, 900, 243] . FixedPtCheck, [567, 324, 25, 525, 100, 675, 375, 900, 243]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y4 + 2 y5, y1, 0, 2 y1 - y2 - y3 + 3 y5, 0, y2, y3, y4, y5]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 + y5, y1, y2, y3, y4, 0, y5, 0, 0]

 

  p = - s 5 + s 6

 » SYNC'D 4545/524288 , 0.008668899536

 
193 . Coloring, {2, 4, 7, 8, 9}

R: [4, 9, 4, 8, 7, 7, 5, 6, 2]    B: [2, 4, 5, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` - 5 + τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 5 + τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-259, -148, -125, -325, -500, -225, -625, -300, -111] . FixedPtCheck, [259, 148, 125, 325, 500, 225, 625, 300, 111]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y5 + y3 + y4 - 4 y1, 0, y5, y3, y4, y2, 4 y5 + 4 y3 + 4 y4 - 15 y1 - y2, y1]

 

  p = s 4 - s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 4, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 - y2 + 2 y3, 2 y1 + 3 y3 - y4 - y5, y1, y2, y3, 0, y4, y5, 0]

 

  p' = s 2 - s 6   p = s 2 - s 6

 » SYNC'D 6423/524288 , 0.01225090027

 
194 . Coloring, {2, 5, 6, 7, 8}

R: [4, 9, 4, 7, 3, 8, 5, 6, 1]    B: [2, 4, 5, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [767, 236, 1161, 1505, 1548, 645, 1677, 860, 177] . FixedPtCheck, [767, 236, 1161, 1505, 1548, 645, 1677, 860, 177]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )`` (` 1 + τ ` )` 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 8 6 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, 0, y6, -y1 + 4 y5 + 4 y6 + 4 y2 - 15 y3 - y4, y1, y5 + y6 + y2 - 4 y3, y2, y3, y4]

 

  p' = s 3 - s 7   p = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y1, y2, 0, y5, y3, 0, y4, y6, 0]  

See Matrices
 

 » SYNC'D 104409/4194304 , 0.02489304543

 
195 . Coloring, {2, 5, 6, 7, 9}

R: [4, 9, 4, 7, 3, 8, 5, 1, 2]    B: [2, 4, 5, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [7, 4, 9, 13, 12, 1, 13, 4, 3] . FixedPtCheck, [7, 4, 9, 13, 12, 1, 13, 4, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[y6, y5, y4, y3, y2, 0, -y6 + 3 y5 - y4 + 2 y1, 2 y5 - y3 - y2 + 3 y1, y1]

 

  p = - s 3 + s 7   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[y3 - y2 - y1 - y6 + y5 + y4, y3, 0, y2, y1, y6, y5, y4, 0]

 

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

 » SYNC'D 47451/4194304 , 0.01131320000

 
196 . Coloring, {2, 5, 6, 8, 9}

R: [4, 9, 4, 7, 3, 8, 1, 6, 2]    B: [2, 4, 5, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 18` (` 5 + τ + τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [329, 188, 75, 350, 100, 150, 325, 200, 141] . FixedPtCheck, [329, 188, 75, 350, 100, 150, 325, 200, 141]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 4 ` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

Omega Rank for R :  cycles: {{1, 4, 7}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[4 y5 - y1 - y4 + 4 y2 - y3, y5, y1, y4, 0, y2, y3, y5, y2]

 

  p = - s 2 - s 3 + s 5 + s 6   p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[2 y1, 9 y1 + 9 y2 - 11 y3 - 2 y4, 0, 2 y2, 7 y1 + 7 y2 - 9 y3, 0, 2 y3, 2 y4, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 8523/4194304 , 0.002032041550

 
197 . Coloring, {2, 5, 7, 8, 9}

R: [4, 9, 4, 7, 3, 7, 5, 6, 2]    B: [2, 4, 5, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )``]`

For τ=1/2, [686, 392, 1350, 1625, 1800, 375, 1950, 500, 294] . FixedPtCheck, [686, 392, 1350, 1625, 1800, 375, 1950, 500, 294]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y5, y4, y3, y2, -15 y5 + 4 y4 - y3 - y2 + 4 y1, y1, 0, -4 y5 + y4 + y1]

 

  p' = s 2 - s 6   p = s 2 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y3, y4, 0, -y3 + y4 - y1 + y2 + y5, y1, 0, y2, y5, 0]

 

  p = s 3 - s 4 + s 5 - s 6

 » SYNC'D 17723/524288 , 0.03380393982

 
198 . Coloring, {2, 6, 7, 8, 9}

R: [4, 9, 4, 7, 7, 8, 5, 6, 2]    B: [2, 4, 5, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-133, -76, -75, -175, -300, -75, -375, -100, -57] . FixedPtCheck, [133, 76, 75, 175, 300, 75, 375, 100, 57]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 7}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y2, 0, y2 - y1 + 3 y3, y1, y3, 3 y2 + y3, y2, y3]

 

  p' = s 4 - s 6   p' = s 3 - s 5   p' = s 2 - s 6   p = s 2 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {3, 5}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y2 - y1 + 2 y5, 2 y2 + 3 y5 - y4 - y3, y2, y1, y5, 0, y4, y3, 0]

 

  p = - s 2 + s 6   p' = s 2 - s 6

 » SYNC'D 13779/4194304 , 0.003285169601

 
199 . Coloring, {3, 4, 5, 6, 7}

R: [4, 4, 5, 8, 3, 8, 5, 1, 1]    B: [2, 9, 4, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-645, -172, -513, -741, -684, -171, -399, -684, -43] . FixedPtCheck, [645, 172, 513, 741, 684, 171, 399, 684, 43]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y3 + y1 - 3 y2 + y6, -2 y1 + 2 y2 - y6 - y5, -y6 - y4, y3, y1, y2, y6, y4, y5]
  p = - s 2 + 6s 4 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 5 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y4, 0, y3, y1, y2, 0, 0, -y4 + 2 y3 - y1 + 2 y2, 0]

 

  p = - s - s 2 + s 4 + s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, 0, y3, 0, 2 y3, y4, 0, y5]

 

  p = - s 4 + s 6

 » SYNC'D 1297/16384 , 0.07916259766

 
200 . Coloring, {3, 4, 5, 6, 8}

R: [4, 4, 5, 8, 3, 8, 1, 6, 1]    B: [2, 9, 4, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 5 + 3τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 54` (` 5 + 3τ ` )`` (` - 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -27` (` 5 + 3τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [390, 104, 114, 399, 152, 513, 266, 684, 26] . FixedPtCheck, [390, 104, 114, 399, 152, 513, 266, 684, 26]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 3 vs 6

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 + 2 y2 - y4, 0, y1, 2 y1 + 3 y2 - y3, y2, y3, 0, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}, {2, 9}},   net cycles: 0 .    order:   2

See Matrix
 

[4 y1, 4 y1 + 2 y2, 0, 2 y1, 8 y1 + 5 y2 - 2 y3, 0, 2 y3, 0, 2 y2]

 

  p = s 2 - s 4   p' = - s 2 + s 4   p' = - s 3 + s 5

 » SYNC'D 155/32768 , 0.004730224609

 
201 . Coloring, {3, 4, 5, 6, 9}

R: [4, 4, 5, 8, 3, 8, 1, 1, 2]    B: [2, 9, 4, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [650, 200, 102, 663, 136, 153, 238, 612, 50] . FixedPtCheck, [650, 200, 102, 663, 136, 153, 238, 612, 50]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y5, y3, -y4 - y5 + 5 y3 + 5 y1 - y2, y1, 0, 0, y2, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 0 .    order:   6

See Matrix
 

[-y1 + 3 y2 + y4 - y3, y1, 0, y2, y4, 2 y2, 3 y2 + y4, 0, y3]

 

  p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 5

 » SYNC'D 2839/131072 , 0.02165985107

 
202 . Coloring, {3, 4, 5, 7, 8}

R: [4, 4, 5, 8, 3, 7, 5, 6, 1]    B: [2, 9, 4, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 3 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , -9` (` - 1 + τ ` )` 4 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-885, -236, -2565, -1482, -3420, -1026, -1995, -1368, -59] . FixedPtCheck, [885, 236, 2565, 1482, 3420, 1026, 1995, 1368, 59]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[y1, 0, y6, y5, y2, y3, y4, y7, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, 0, y3, 0, 0, y4, y3, y5]

 

  p = - s 4 + s 6

 » SYNC'D 1711/16384 , 0.1044311523

 
203 . Coloring, {3, 4, 5, 7, 9}

R: [4, 4, 5, 8, 3, 7, 5, 1, 2]    B: [2, 9, 4, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 - 3τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 - 3τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [403, 124, 459, 510, 612, 102, 357, 408, 31] . FixedPtCheck, [403, 124, 459, 510, 612, 102, 357, 408, 31]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 0 .    order:   6

See Matrix
 

[4 y1, %1, 4 y2, 4 y4, 4 y5, 0, %1, 4 y3, 0] %1 := 7 y1 - 11 y2 + 7 y4 - 11 y5 + 7 y3

 

  p = s 2 - s 4 - s 5 + s 7   p' = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y4, y5, 0, y6, 0, y1, -y4 - y5 - y6 + 5 y1 + 5 y2 - y3, y2, y3]

 

  p = s 3 + s 4 - s 6 - s 7

 » SYNC'D 256761/4194304 , 0.06121659279

 
204 . Coloring, {3, 4, 5, 8, 9}

R: [4, 4, 5, 8, 3, 7, 1, 6, 2]    B: [2, 9, 4, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [611, 188, 255, 663, 340, 459, 595, 612, 47] . FixedPtCheck, [611, 188, 255, 663, 340, 459, 595, 612, 47]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 8 3 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 6, 7, 8}},   net cycles: 1 .   

See Matrix
 

[y6, y5, y4, y3, y2, y1, -y6 - y5 + 5 y4 - y3 + 5 y2 - y1 - y7, y7, 0]

 

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 9}, {5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y3, y3, 0, y2, y1, 0, 3 y3 - y2 - y1, y2, y3 - y2]

 

  p' = s 4 - s 6   p' = s 3 - s 5   p' = s 2 - s 6   p = s 2 - s 6

 » SYNC'D 604887/33554432 , 0.01802703738

 
205 . Coloring, {3, 4, 6, 7, 8}

R: [4, 4, 5, 8, 7, 8, 5, 6, 1]    B: [2, 9, 4, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-375, -100, -171, -399, -684, -513, -741, -684, -25] . FixedPtCheck, [375, 100, 171, 399, 684, 513, 741, 684, 25]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y1 - y2 - y3 - y5 - y6, y5, y4, y1, y2, y3, -2 y4 - 2 y2 - 2 y3 - y5 - y6, y4 + 2 y2 + 2 y3 + y5 + y6, y6]
  p = - s 3 + s 4 + 4s 5 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, 0, 3 y1 - y3 - 4 y2 + 3 y4, 2 y1 - 3 y2 + 2 y4, y3, y2, y4, 0]

 

  p = s 3 - s 5   p' = s 3 - s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   6

[y2, y1, y3, y4, 0, 0, y5, 0, y6]  

See Matrices
 

 » SYNC'D 329/8192 , 0.04016113281

 
206 . Coloring, {3, 4, 6, 7, 9}

R: [4, 4, 5, 8, 7, 8, 5, 1, 2]    B: [2, 9, 4, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [-637, -196, -153, -663, -612, -153, -663, -612, -49] . FixedPtCheck, [637, 196, 153, 663, 612, 153, 663, 612, 49]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[2 y5 + y3 + y2, -y5 - y3 - y2 - y1 - y4, -y5 - y3, y1, -y5 - y2, y4, y5, y3, y2]
  p = s 2 - 2s 3 + 4s 4 - 4s 5 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s 2 + 4s 4 + 4s 5 + 8s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 7 8 vs 8 8 vs 8 5 vs 6 4 vs 7

Omega Rank for R :  cycles: {{1, 4, 8}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[-y4 - y3 + 2 y2 + 2 y1 - y5, y4, 0, y3, y2, 0, y1, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y4, y3, y2, y4 - y3, 0, y2, y1, 0, y4 + y2 - y1]

 

  p' = s 4 - s 6   p' = s 5 - s 6   p = s 4 - s 7

 » SYNC'D 6495/131072 , 0.04955291748

 
207 . Coloring, {3, 4, 6, 8, 9}

R: [4, 4, 5, 8, 7, 8, 1, 6, 2]    B: [2, 9, 4, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 3τ 2 ` )` , -54` (` - 1 + τ ` )` 2 ` (` 5 + 3τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )` 3 ` (` 5 + 3τ 2 ` )``]`

For τ=1/2, [-598, -184, -34, -595, -136, -765, -442, -1020, -46] . FixedPtCheck, [598, 184, 34, 595, 136, 765, 442, 1020, 46]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   6

See Matrix
 

[y2, y3, 0, y4, y3, y2 - y4 + 2 y3 - y1 + y5, y1, y5, 0]

 

  p = - s 5 + s 6   p = - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7}, {1, 2, 9}},   net cycles: 2 .   

See Matrix
 

[4 y6, 4 y5, 4 y4, 4 y3, 4 y2, 0, 5 y6 + 5 y5 - 4 y4 - 4 y3 - 4 y2 + 5 y1, 0, 4 y1]

 

  p = - s - s 2 - s 3 + s 5 + s 6 + s 7

 » SYNC'D 48415/4194304 , 0.01154303551

 
208 . Coloring, {3, 4, 7, 8, 9}

R: [4, 4, 5, 8, 7, 7, 5, 6, 2]    B: [2, 9, 4, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ 2 + τ 4 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ 2 + τ 4 ` )` , 9` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ 2 + τ 4 ` )``]`

For τ=1/2, [-1157, -356, -765, -1326, -3060, -918, -3315, -1224, -89] . FixedPtCheck, [1157, 356, 765, 1326, 3060, 918, 3315, 1224, 89]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

[0, y1, 0, y2, y6, y3, y4, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y3, y1, 2 y5, y2, 0, 0, y4, y5, y6]

 

  p = - s 4 + s 7

 » SYNC'D 7785/65536 , 0.1187896729

 
209 . Coloring, {3, 5, 6, 7, 8}

R: [4, 4, 5, 7, 3, 8, 5, 6, 1]    B: [2, 9, 4, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 18` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , 9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 4 , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , -18` (` - 5 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [-435, -116, -1539, -798, -2052, -342, -1197, -456, -29] . FixedPtCheck, [435, 116, 1539, 798, 2052, 342, 1197, 456, 29]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y4, 0, -y4 + y2 - y3 + 4 y5, y1, 4 y2 + y5 - y1, y2, y3, y5, 0]

 

  p = s 4 - s 6   p' = s 4 - s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: -1 .    order:   4

See Matrix
 

[y2, y1, 0, y3, 0, 0, 3 y3, y5, y4]

 

  p = - s 4 + s 6

 » SYNC'D 141/4096 , 0.03442382812

 
210 . Coloring, {3, 5, 6, 7, 9}

R: [4, 4, 5, 7, 3, 8, 5, 1, 2]    B: [2, 9, 4, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )` 3 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 18` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 4 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [767, 236, 2295, 1326, 3060, 102, 1785, 408, 59] . FixedPtCheck, [767, 236, 2295, 1326, 3060, 102, 1785, 408, 59]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, y5, y2, y3, y4, 0, y6, y5, 0]

 

  p = s 5 - s 7

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y7, y6, 0, y5, 0, y3, y4, y2, y1]  

See Matrices
 

 » SYNC'D 52455/2097152 , 0.02501249313

 
211 . Coloring, {3, 5, 6, 8, 9}

R: [4, 4, 5, 7, 3, 8, 1, 6, 2]    B: [2, 9, 4, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 153, 357, 204, 153, 357, 204, 25] . FixedPtCheck, [325, 100, 153, 357, 204, 153, 357, 204, 25]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 9 9 vs 9 5 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 7}, {6, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[y1, -y1 + 4 y2 - y4 + 4 y5 - y3, y2, y4, y5, y2, y3, y5, 0]

 

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}, {5, 7}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, -y5 - y1 + 4 y2 - y3 - y4, 0, y1, y2, 0, y2, y3, y4]

 

  p = - s 3 + s 6   p' = - s 3 + s 6

 » SYNC'D 867735/134217728 , 0.006465129554

 
212 . Coloring, {3, 5, 7, 8, 9}

R: [4, 4, 5, 7, 3, 7, 5, 6, 2]    B: [2, 9, 4, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , -18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )` 4 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [325, 100, 1377, 663, 1836, 153, 1071, 204, 25] . FixedPtCheck, [325, 100, 1377, 663, 1836, 153, 1071, 204, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y5, y4, y3, y2, 2 y5, y1, 0, 0]

 

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y3, 0, y1, 0, 0, 2 y1, y4, y5]

 

  p = - s 3 + s 6

 » SYNC'D 933/8192 , 0.1138916016

 
213 . Coloring, {3, 6, 7, 8, 9}

R: [4, 4, 5, 7, 7, 8, 5, 6, 2]    B: [2, 9, 4, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` 5 + τ + τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-611, -188, -459, -714, -1836, -306, -1989, -408, -47] . FixedPtCheck, [611, 188, 459, 714, 1836, 306, 1989, 408, 47]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y2 - y3 + 4 y4, 0, -y1 + 4 y2 + y4, y1, y2, y3, y4, 0]

 

  p' = s 3 - s 5   p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   6

See Matrix
 

[y1, y2, 2 y4, y3, 0, 0, y4, y5, y6]

 

  p = - s 4 + s 7

 » SYNC'D 7935/131072 , 0.06053924561

 
214 . Coloring, {4, 5, 6, 7, 8}

R: [4, 4, 4, 8, 3, 8, 5, 6, 1]    B: [2, 9, 5, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )` 3 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 3 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` - 5 + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-1290, -344, -1026, -1995, -1368, -2565, -1482, -3420, -86] . FixedPtCheck, [1290, 344, 1026, 1995, 1368, 2565, 1482, 3420, 86]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, y2, y3, 3 y1, y4, 0, y5, 0]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   4

[y1, y5, 0, 0, y3, 0, y4, 0, y2]  

See Matrices
 

 » SYNC'D 25/256 , 0.09765625000

 
215 . Coloring, {4, 5, 6, 7, 9}

R: [4, 4, 4, 8, 3, 8, 5, 1, 2]    B: [2, 9, 5, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )`` (` 3 + τ 2 ` )` , -54` (` - 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )` 2 ` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [2678, 824, 918, 3315, 1224, 765, 1326, 3060, 206] . FixedPtCheck, [2678, 824, 918, 3315, 1224, 765, 1326, 3060, 206]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 4, 8}},   net cycles: -1 .    order:   3

See Matrix
 

[y4, y5, y2, y3, 3 y5, 0, 0, y1, 0]

 

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y3, 0, 0, y1, 2 y1, y4, 0, y5]

 

  p = - s 3 + s 6

 » SYNC'D 4293/65536 , 0.06550598145

 
216 . Coloring, {4, 5, 6, 8, 9}

R: [4, 4, 4, 8, 3, 8, 1, 6, 2]    B: [2, 9, 5, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-325, -100, -51, -357, -68, -459, -221, -612, -25] . FixedPtCheck, [325, 100, 51, 357, 68, 459, 221, 612, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 4 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -2 .    order:   4

See Matrix
 

[3 y3, y3, 2 y3, y2, 0, y1, 0, y4, 0]

 

  p' = - s 3 + s 5   p = s 3 - s 5

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 2 .    order:   6

See Matrix
 

[4 y3, 4 y2, 0, 0, 4 y1, 0, 5 y3 + 5 y2 - 4 y1 + 5 y4, 0, 4 y4]

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 25/1024 , 0.02441406250

 
217 . Coloring, {4, 5, 7, 8, 9}

R: [4, 4, 4, 8, 3, 7, 5, 6, 2]    B: [2, 9, 5, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 459, 663, 612, 459, 663, 612, 25] . FixedPtCheck, [325, 100, 459, 663, 612, 459, 663, 612, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 7 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 8}},   net cycles: 0 .    order:   6

[0, y2, y1, y3, y4, y5, y6, y7, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, 0, y4, 0, y5, y4, y3]

 

  p = - s 3 + s 6

 » SYNC'D 298755/4194304 , 0.07122874260

 
218 . Coloring, {4, 6, 7, 8, 9}

R: [4, 4, 4, 8, 7, 8, 5, 6, 2]    B: [2, 9, 5, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 + τ ` )`` (` - 1 + τ ` )` 2 , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + τ ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-286, -88, -102, -357, -408, -459, -510, -612, -22] . FixedPtCheck, [286, 88, 102, 357, 408, 459, 510, 612, 22]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1, 0, y4, -5 y1 + 14 y2 - 5 y3, -14 y1 - y4 + 39 y2 - 14 y3, y2, y3, 0]

 

  p' = - s 3 + s 5   p = s 3 - s 5

Omega Rank for B :  cycles: {{3, 5}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[y5, y4, y3, 0, y2, 0, y1, 0, -y5 - y4 + 5 y3 + 5 y2 - y1]

 

  p = s 2 + s 3 - s 5 - s 6

 » SYNC'D 3125/131072 , 0.02384185791

 
219 . Coloring, {5, 6, 7, 8, 9}

R: [4, 4, 4, 7, 3, 8, 5, 6, 2]    B: [2, 9, 5, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 4 , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 18` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [767, 236, 1377, 1785, 1836, 765, 1989, 1020, 59] . FixedPtCheck, [767, 236, 1377, 1785, 1836, 765, 1989, 1020, 59]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, -y1 + y2 - y3 + 4 y4, y1, 4 y2 + y4 - y5, y5, y2, y3, y4, 0]

 

  p = - s 2 + s 6   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y4, 0, 0, y3, 0, y2, 3 y3, y5]

 

  p = - s 3 + s 6

 » SYNC'D 4347/65536 , 0.06632995605

 
220 . Coloring, {2, 3, 4, 5, 6, 7}

R: [4, 9, 5, 8, 3, 8, 5, 1, 1]    B: [2, 4, 4, 7, 7, 7, 1, 6, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [13, 4, 9, 13, 12, 3, 7, 12, 3] . FixedPtCheck, [13, 4, 9, 13, 12, 3, 7, 12, 3]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y2 - 3 y3 + y4 - 2 y6 - y1, -2 y2 + 2 y3 - y4 + y6, -y4 - y5, y1, y2, y3, y4, y5, y6]
  p = s 3 + s 4 - 4s 5 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 5

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y3, 0, y4, y5, y2, 0, 0, -y3 + 2 y4 - y5 + 2 y2 - y1, y1]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

[y1, y2, 0, y3, 0, y4, y5, 0, 0]  

See Matrices
 

 » SYNC'D 509/8192 , 0.06213378906

 
221 . Coloring, {2, 3, 4, 5, 6, 8}

R: [4, 9, 5, 8, 3, 8, 1, 6, 1]    B: [2, 4, 4, 7, 7, 7, 5, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-27` (` 5 + 3τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 54` (` 5 + 3τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 27` (` 5 + 3τ ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 2 `]`

For τ=1/2, [1014, 312, 258, 903, 344, 1161, 602, 1548, 234] . FixedPtCheck, [1014, 312, 258, 903, 344, 1161, 602, 1548, 234]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 5

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[3 y1 + 2 y2 - y4, 0, y1, 2 y1 + 3 y2 - y3 - y5, y2, y3, 0, y4, y5]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

[y1, y2, 0, y3, y4, 0, y5, 0, 0]  

See Matrices
 

 » SYNC'D 579/32768 , 0.01766967773

 
222 . Coloring, {2, 3, 4, 5, 6, 9}

R: [4, 9, 5, 8, 3, 8, 1, 1, 2]    B: [2, 4, 4, 7, 7, 7, 5, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )` , 18` (` 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 `]`

For τ=1/2, [42, 24, 6, 39, 8, 9, 14, 36, 18] . FixedPtCheck, [42, 24, 6, 39, 8, 9, 14, 36, 18]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {1, 4, 8}},   net cycles: 3 .    order:   6

See Matrix
 

[4 y2 + 4 y4 - y1 - y3, y2, y4, y1, y2, 0, 0, y3, y4]

 

  p' = s + s 2 - s 4 - s 5   p = - s - s 2 + s 4 + s 5   p = - s + s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 3 y1 + y3 + y2 - y4, 0, y3, y2, 2 y1, y4, 0, 0]

 

  p' = s 4 - s 5   p = s 4 - s 6

 » SYNC'D 28791/2097152 , 0.01372861862

 
223 . Coloring, {2, 3, 4, 5, 7, 8}

R: [4, 9, 5, 8, 3, 7, 5, 6, 1]    B: [2, 4, 4, 7, 7, 8, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 3 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-767, -236, -1935, -1118, -2580, -774, -1505, -1032, -177] . FixedPtCheck, [767, 236, 1935, 1118, 2580, 774, 1505, 1032, 177]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 8 vs 8 5 vs 5

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   8

[y3, 0, y1, y2, y6, y7, y8, y4, y5]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

[y5, y4, 0, y3, 0, 0, y2, y1, 0]  

See Matrices
 

 » SYNC'D 45091/1048576 , 0.04300212860

 
224 . Coloring, {2, 3, 4, 5, 7, 9}

R: [4, 9, 5, 8, 3, 7, 5, 1, 2]    B: [2, 4, 4, 7, 7, 8, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` 5 - 3τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - 3τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [217, 124, 225, 250, 300, 50, 175, 200, 93] . FixedPtCheck, [217, 124, 225, 250, 300, 50, 175, 200, 93]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 8 4 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {1, 4, 8}},   net cycles: 2 .    order:   6

See Matrix
 

[-2 y5 - 2 y3 - 8 y4 + 8 y1, 3 y1 - 5 y4, -2 y2 - 7 y4 + 5 y1, 2 y5, 2 y1, 0, 2 y2, 2 y3, 2 y4]

 

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {6, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[-y3 + 2 y2 + 3 y1, y4, 0, y3, 0, y2, -y4 + 3 y2 + 2 y1, y1, 0]

 

  p = - s + s 5   p' = s - s 5

 » SYNC'D 84291/16777216 , 0.005024135113

 
225 . Coloring, {2, 3, 4, 5, 8, 9}

R: [4, 9, 5, 8, 3, 7, 1, 6, 2]    B: [2, 4, 4, 7, 7, 8, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [987, 564, 375, 975, 500, 675, 875, 900, 423] . FixedPtCheck, [987, 564, 375, 975, 500, 675, 875, 900, 423]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 8 vs 9 6 vs 9 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {1, 4, 6, 7, 8}},   net cycles: 3 .   

See Matrix
 

[4 y2 - y1 + 4 y6 - y5 - y4 - y3, y6, y2, y1, y6, y5, y4, y3, y2]

 

  p' = s 2 + s 3 - s 7 - s 8   p' = 1 + s 3 - s 5 - s 8   p' = s - s 3 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   6

See Matrix
 

[y1 - y2 - y3 + y4 + y5, y1, 0, y2, y3, 0, y4, y5, 0]

 

  p = - s 5 + s 6

 » SYNC'D 399/32768 , 0.01217651367

 
226 . Coloring, {2, 3, 4, 6, 7, 8}

R: [4, 9, 5, 8, 7, 8, 5, 6, 1]    B: [2, 4, 4, 7, 3, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [325, 100, 129, 301, 516, 387, 559, 516, 75] . FixedPtCheck, [325, 100, 129, 301, 516, 387, 559, 516, 75]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[-y4 - 2 y5 + y3 + y1 - 2 y6 - y2, y4 + 2 y5 - 2 y3 - 2 y1 + y6, -y4 - y5, y2, y3, y1, y4, y5, y6]
  p = s 2 - 2s 4 - 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 7 5 vs 5

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y1, 0, 0, 3 y1 - y3 - 4 y4 + 3 y2 - y5, 2 y1 - 3 y4 + 2 y2, y3, y4, y2, y5]

 

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

[y3, y1, y2, y4, 0, 0, y5, 0, 0]  

See Matrices
 

 » SYNC'D 111/4096 , 0.02709960938

 
227 . Coloring, {2, 3, 4, 6, 7, 9}

R: [4, 9, 5, 8, 7, 8, 5, 1, 2]    B: [2, 4, 4, 7, 3, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 18` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , -9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 5 + 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-343, -196, -75, -325, -300, -75, -325, -300, -147] . FixedPtCheck, [343, 196, 75, 325, 300, 75, 325, 300, 147]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {1, 4, 8}, {2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[3 y1 - y2 - y3 + 3 y4, y1, 0, y2, 2 y4, 0, 2 y1, y3, y4]

 

  p' = - s - s 2 + s 4 + s 5   p = - s + s 7   p = - s + s 3 + s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y2, y4, -y1 + y2 + y3, 0, y4, y3, 0, 0]

 

  p' = s 2 - s 3 + s 4 - s 5   p = s 2 - s 6

 » SYNC'D 16725/2097152 , 0.007975101471

 
228 . Coloring, {2, 3, 4, 6, 8, 9}

R: [4, 9, 5, 8, 7, 8, 1, 6, 2]    B: [2, 4, 4, 7, 3, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ ` )`` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 54` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 5 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 `]`

For τ=1/2, [-966, -552, -50, -875, -200, -1125, -650, -1500, -414] . FixedPtCheck, [966, 552, 50, 875, 200, 1125, 650, 1500, 414]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 8 5 vs 6

Omega Rank for R :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[2 y1 - y2 - y5 + 3 y6, y1, 0, 3 y1 - y3 - y4 + 2 y6, y2, y3, y4, y5, y6]

 

  p = - s 5 + s 7   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y3 + y4 - y5 - y1 + y2, y3, y4, y5, y1, 0, y2, 0, 0]

 

  p = - s 3 + s 4 - s 5 + s 6

 » SYNC'D 69969/4194304 , 0.01668190956

 
229 . Coloring, {2, 3, 4, 7, 8, 9}

R: [4, 9, 5, 8, 7, 7, 5, 6, 2]    B: [2, 4, 4, 7, 3, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 2τ 2 + τ 4 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 5 + 2τ 2 + τ 4 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + 2τ 2 + τ 4 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-623, -356, -375, -650, -1500, -450, -1625, -600, -267] . FixedPtCheck, [623, 356, 375, 650, 1500, 450, 1625, 600, 267]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 9}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1 + y3 + y2 - 4 y4, 0, y1, y3, y2, 4 y1 + 4 y3 + 4 y2 - 15 y4 - y5, y5, y4]

 

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, y1 + y2 - y3 - 3 y4, 2 y4, y2, 0, 0, y3, y4, 0]

 

  p = s 2 - s 6   p' = s 2 - s 3 + s 4 - s 5

 » SYNC'D 16285/524288 , 0.03106117249

 
230 . Coloring, {2, 3, 5, 6, 7, 8}

R: [4, 9, 5, 7, 3, 8, 5, 6, 1]    B: [2, 4, 4, 8, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` 5 + 4τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 3 ` (` 5 + 4τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 3 ` (` 5 + 4τ + τ 2 ` )``]`

For τ=1/2, [-377, -116, -1161, -602, -1548, -258, -903, -344, -87] . FixedPtCheck, [377, 116, 1161, 602, 1548, 258, 903, 344, 87]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 6 vs 8 5 vs 5

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + y4 - y5 + 4 y6, 0, y1, -y3 + 4 y4 + y6 - y2, y3, y4, y5, y6, y2]

 

  p = s 5 - s 7   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

[y1, y2, 0, y4, 0, 0, y3, y5, 0]  

See Matrices
 

 » SYNC'D 18381/1048576 , 0.01752948761

 
231 . Coloring, {2, 3, 5, 6, 7, 9}

R: [4, 9, 5, 7, 3, 8, 5, 1, 2]    B: [2, 4, 4, 8, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [413, 236, 1125, 650, 1500, 50, 875, 200, 177] . FixedPtCheck, [413, 236, 1125, 650, 1500, 50, 875, 200, 177]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 8 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[y2, y3, -y2 + 3 y3 - y4 + 2 y6, 2 y3 - y1 - y5 + 3 y6, y1, 0, y4, y5, y6]

 

  p = s 5 - s 7   p' = s 5 - s 7

Omega Rank for B :  cycles: {{1, 2, 4, 6, 7, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[y1 - y2 - y5 + y3 + y4, y1, 0, y2, 0, y5, y3, y4, 0]

 

  p = s - s 2 + s 3 - s 4 + s 5 - s 6

 » SYNC'D 365379/33554432 , 0.01088914275

 
232 . Coloring, {2, 3, 5, 6, 8, 9}

R: [4, 9, 5, 7, 3, 8, 1, 6, 2]    B: [2, 4, 4, 8, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `3` (` 3 + τ ` )` , 6 , 3` (` 1 + τ ` )` , 3` (` 3 + τ ` )` , 6 , 3` (` 1 + τ ` )` , 3` (` 3 + τ ` )` , 6 , 3` (` 1 + τ ` )``]`

For τ=1/2, [7, 4, 3, 7, 4, 3, 7, 4, 3] . FixedPtCheck, [7, 4, 3, 7, 4, 3, 7, 4, 3]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 3

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[0, -y1, y1, 0, -y1, y1, 0, -y1, y1]
  p = s - 64s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true

  p' = s + 32s 6   p' = s 2 - 16s 6   p' = s 3 + 8s 6   p' = s 4 - 4s 6   p' = s 5 + 2s 6
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 7 2 vs 9 2 vs 9 2 vs 9 1 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 7}, {2, 9}, {6, 8}},   net cycles: 4 .    order:   6

See Matrix
 

[y1 + y2, y1, y2, y1 + y2, y1, y2, y1 + y2, y1, y2]

 

  p' = - 1 + s 2   p' = - s + s 3   p' = - 1 + s 4   p' = - s + s 5   p' = - 1 + s 6   p' = - s + s 7   p' = - 1 + s 8

Omega Rank for B :  cycles: {{1, 2, 4, 8}, {5, 7}},   net cycles: 2 .    order:   4

See Matrix
 

[y1, y1, 0, y1, y1, 0, y1, y1, 0]

 

  p = - s + s 4   p = - s + s 6   p = - s + s 5   p = - s + s 3   p = - s + s 2


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, x4, x5, x6, -64 x1 + 32 x2 - 16 x3 + 8 x4 - 4 x5 + 2 x6]
For A+2Δ :   [y4, y2, y3, y1, -y4 - y2 - y1 - y5 - y6, -y4 - y3 - y1 - y5 - y7, y5, y6, y7]
For A-2Δ :   [-y2 - y1 - y3 - y4 - y6, y2, y2 + y3 + y6 - y5 - y7, y1, y3, y5, y4, y6, y7]

Range of {ΩΔi}: [0, -μ1, μ1, 0, -μ1, μ1, 0, -μ1, μ1]

 
rank of M is 9 , rank of N is 8

M              \ ;    N

$ [ [0, 8, 4, 15, 8, 4, 15, 8, 4] , [8, 0, 0, 8, 10, 0, 8, 10, 0] , [4, 0, 0, 4, 0, 5, 4, 0, 5] , [15, 8, 4, 0, 8, 4, 15, 8, 4] , [8, 10, 0, 8, 0, 0, 8, 10, 0] , [4, 0, 5, 4, 0, 0, 4, 0, 5] , [15, 8, 4, 15, 8, 4, 0, 8, 4] , [8, 10, 0, 8, 10, 0, 8, 0, 0] , [4, 0, 5, 4, 0, 5, 4, 0, 0] ] $     $ [ [0, 3, 3, 3, 3, 3, 3, 3, 3] , [3, 0, 1, 3, 3, 2, 3, 3, 3] , [3, 1, 0, 3, 3, 3, 3, 2, 3] , [3, 3, 3, 0, 3, 3, 3, 3, 3] , [3, 3, 3, 3, 0, 1, 3, 3, 2] , [3, 2, 3, 3, 1, 0, 3, 3, 3] , [3, 3, 3, 3, 3, 3, 0, 3, 3] , [3, 3, 2, 3, 3, 3, 3, 0, 1] , [3, 3, 3, 3, 2, 3, 3, 1, 0] ] $

Check is ΩΔN zero? true, πΔ= [0, -1, 1, 0, -1, 1, 0, -1, 1]

ker M, [0, 0, 0, 0, 0, 0, 0, 0, 0]
Range M, [x5, x1, x2, x3, x4, x6, x7, x8, x9]

τ= 15 , r'= 5/6

Ranges

Action of R on ranges, [[2], [1]]
Action of B on ranges, [[1], [1]]
β({1, 2, 4, 5, 7, 8}) = 2/3
β({1, 3, 4, 6, 7, 9}) = 1/3

ker N, [0, μ1, -μ1, 0, μ1, -μ1, 0, μ1, -μ1]
Range of N
    [y1, y7 - y6 + y5 - y3 + y2, y7, y8, y6, y5, y4, y3, y2]

Partitions

Action of R on partitions, [[2], [1]]
Action of B on partitions, [[1], [1]]

α([{1}, {4}, {2, 3}, {8, 9}, {7}, {5, 6}]) = 2/3
α([{1}, {4}, {5, 9}, {2, 6}, {7}, {3, 8}]) = 1/3

b1 = {1} ` , ` b2 = {4} ` , ` b3 = {2, 3} ` , ` b4 = {8, 9} ` , ` b5 = {5, 9} ` , ` b6 = {2, 6} ` , ` b7 = {7} ` , ` b8 = {3, 8} ` , ` b9 = {5, 6}

Action of R and B on the blocks of the partitions: = [7, 1, 5, 6, 3, 4, 2, 9, 8] [4, 3, 1, 2, 7, 1, 9, 2, 7]
with invariant measure [3, 3, 2, 2, 1, 1, 3, 1, 2]

N by blocks, check: true . ` See partition graph.

` ` See level-6 partition graph.

`

Sandwich
Coloring {2, 3, 5, 6, 8, 9}
Rank6
R,B [4, 9, 5, 7, 3, 8, 1, 6, 2], [2, 4, 4, 8, 7, 7, 5, 1, 1]
π2 [8, 4, 15, 8, 4, 15, 8, 4, 0, 8, 10, 0, 8, 10, 0, 4, 0, 5, 4, 0, 5, 8, 4, 15, 8, 4, 0, 8, 10, 0, 4, 0, 5, 8, 4, 0]
u2 [3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 3, 3, 3, 1] (dim 2)
wpp [1, 2, 2, 1, 2, 2, 1, 2, 2]
π6 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u6 [0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
233 . Coloring, {2, 3, 5, 7, 8, 9}

R: [4, 9, 5, 7, 3, 7, 5, 6, 2]    B: [2, 4, 4, 8, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 , 18` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [7, 4, 27, 13, 36, 3, 21, 4, 3] . FixedPtCheck, [7, 4, 27, 13, 36, 3, 21, 4, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R :  cycles: {{3, 5}, {2, 9}},   net cycles: 0 .    order:   4

See Matrix
 

[0, 2 y2 + 5 y1 - 8 y3, -2 y4 - 30 y3 + 8 y2 + 20 y1, 3 y1, 2 y2, 2 y1, 2 y4, 0, 2 y3]

 

  p = - s 3 + s 5   p = - s 3 + s 7   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 - y2 + y4 + y3, y1, 0, y2, 0, 0, y4, y3, 0]

 

  p = s 2 - s 3 + s 4 - s 5

 » SYNC'D 481/16384 , 0.02935791016

 
234 . Coloring, {2, 3, 6, 7, 8, 9}

R: [4, 9, 5, 7, 7, 8, 5, 6, 2]    B: [2, 4, 4, 8, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 18` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + τ + τ 2 + τ 3 ` )``]`

For τ=1/2, [-329, -188, -225, -350, -900, -150, -975, -200, -141] . FixedPtCheck, [329, 188, 225, 350, 900, 150, 975, 200, 141]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` τ ` )` 2 ` (` - 1 + τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 3 vs 7 4 vs 6

Omega Rank for R :  cycles: {{5, 7}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y2, 0, y2 - y1 + 3 y3, y1, y3, 3 y2 + y3, y2, y3]

 

  p' = s 3 - s 5   p' = s 4 - s 6   p = s 2 - s 6   p' = s 2 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y4, y3, 2 y1, y2, 0, 0, y1, y4 - y3 + y2 - 3 y1, 0]

 

  p = - s 2 + s 3 - s 4 + s 5   p = - s 2 + s 6

 » SYNC'D 29799/4194304 , 0.007104635239

 
235 . Coloring, {2, 4, 5, 6, 7, 8}

R: [4, 9, 4, 8, 3, 8, 5, 6, 1]    B: [2, 4, 5, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` - 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 1 + τ 2 ` )` , 9` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )``]`

For τ=1/2, [26, 8, 18, 35, 24, 45, 26, 60, 6] . FixedPtCheck, [26, 8, 18, 35, 24, 45, 26, 60, 6]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 7 5 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[y1, 0, y2, y3, -9 y1 + 6 y2, y4, 0, y5, -6 y1 + 4 y2]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

[y4, y2, 0, y3, y1, 0, y5, 0, 0]  

See Matrices
 

 » SYNC'D 365/8192 , 0.04455566406

 
236 . Coloring, {2, 4, 5, 6, 7, 9}

R: [4, 9, 4, 8, 3, 8, 5, 1, 2]    B: [2, 4, 5, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` 3 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , 54` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 27` (` 1 + τ ` )`` (`5 - 2τ + 8τ 2 + 2τ 3 + 3τ 4 ` )``]`

For τ=1/2, [1442, 824, 450, 1625, 600, 375, 650, 1500, 618] . FixedPtCheck, [1442, 824, 450, 1625, 600, 375, 650, 1500, 618]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 6 vs 7 4 vs 6

Omega Rank for R :  cycles: {{2, 9}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[5 y1 - y2 - y3 - y4 - y5 + 5 y6, y1, y2, y3, y4, 0, 0, y5, y6]

 

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1 - y3 - 3 y4 + y2, y1, 0, y3, y4, 2 y4, y2, 0, 0]

 

  p = - s 2 + s 6   p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 4543/131072 , 0.03466033936

 
237 . Coloring, {2, 4, 5, 6, 8, 9}

R: [4, 9, 4, 8, 3, 8, 1, 6, 2]    B: [2, 4, 5, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` - 1 + τ ` )` 2 , -9` (` 1 + τ ` )` 3 , 9` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )``]`

For τ=1/2, [-21, -12, -3, -21, -4, -27, -13, -36, -9] . FixedPtCheck, [21, 12, 3, 21, 4, 27, 13, 36, 9]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R :  cycles: {{2, 9}, {6, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[3 y1, 5 y1 + 2 y3 - 8 y4, 2 y1, 20 y1 + 8 y3 - 30 y4 - 2 y2, 0, 2 y2, 0, 2 y3, 2 y4]

 

  p' = s 4 - s 6   p = s 3 - s 7   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 - y2 - y3 + y4, y1, 0, y2, y3, 0, y4, 0, 0]

 

  p = - s 4 + s 5

 » SYNC'D 1771/65536 , 0.02702331543

 
238 . Coloring, {2, 4, 5, 7, 8, 9}

R: [4, 9, 4, 8, 3, 7, 5, 6, 2]    B: [2, 4, 5, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` , 9` (` 1 + τ ` )` 2 , 9` (` 3 + τ 2 ` )` , 18` (` 1 + τ ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [7, 4, 9, 13, 12, 9, 13, 12, 3] . FixedPtCheck, [7, 4, 9, 13, 12, 9, 13, 12, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 7 vs 8 6 vs 8 4 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 8}, {2, 9}},   net cycles: 2 .    order:   6

See Matrix
 

[0, y1 + y5 + y4 - 4 y6, 4 y1 + 4 y5 + 4 y4 - 15 y6 - y3 - y2, y1, y5, y4, y3, y2, y6]

 

  p = s - s 7   p' = s - s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: -1 .    order:   4

See Matrix
 

[y1 - y2 + y3, y1, 0, y2, y4, 0, y3, y4, 0]

 

  p = - s 2 + s 3 - s 4 + s 5   p = - s 2 + s 6

 » SYNC'D 1431675/33554432 , 0.04266723990

 
239 . Coloring, {2, 4, 6, 7, 8, 9}

R: [4, 9, 4, 8, 7, 8, 5, 6, 2]    B: [2, 4, 5, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )` , 18` (` 5 + τ ` )`` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-154, -88, -50, -175, -200, -225, -250, -300, -66] . FixedPtCheck, [154, 88, 50, 175, 200, 225, 250, 300, 66]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 3 vs 7 4 vs 6

Omega Rank for R :  cycles: {{2, 9}, {6, 8}, {5, 7}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y1, 0, -10 y1 - y2 + 8 y3, -5 y1 + 4 y3, y2, y3, 2 y3, -4 y1 + 3 y3]

 

  p = - s 2 + s 4   p' = - s 2 + s 4   p = - s 2 + s 6   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}, {3, 5}},   net cycles: 2 .    order:   4

See Matrix
 

[3 y1 - y3 + 2 y2, 2 y1 + 3 y2 - y4, y1, y3, y2, 0, y4, 0, 0]

 

  p = - s + s 5   p' = - s + s 5

 » SYNC'D 447/262144 , 0.001705169678

 
240 . Coloring, {2, 5, 6, 7, 8, 9}

R: [4, 9, 4, 7, 3, 8, 5, 6, 2]    B: [2, 4, 5, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [413, 236, 675, 875, 900, 375, 975, 500, 177] . FixedPtCheck, [413, 236, 675, 875, 900, 375, 975, 500, 177]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 7 vs 8 4 vs 8 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 5, 7}, {2, 9}, {6, 8}},   net cycles: 3 .    order:   4

See Matrix
 

[0, y3, 3 y3 + y4 - y2, y1, y3 - y1 + 3 y4, y4, y2, y3, y4]

 

  p' = - s 2 + s 6   p' = s - s 5   p = s - s 5   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 - y2 - y3 + y4 + y5, y1, 0, y2, y3, 0, y4, y5, 0]

 

  p = - s 3 + s 4 - s 5 + s 6

 » SYNC'D 469899/33554432 , 0.01400408149

 
241 . Coloring, {3, 4, 5, 6, 7, 8}

R: [4, 4, 5, 8, 3, 8, 5, 6, 1]    B: [2, 9, 4, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 18` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 , -9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 , -9` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [15, 4, 27, 21, 36, 27, 21, 36, 1] . FixedPtCheck, [15, 4, 27, 21, 36, 27, 21, 36, 1]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y5 - y2 + y1 + y4, -y5 - 2 y1 - 2 y4 - y6, -y5 - y3, y2, y1, y4, y5, y3, y6]
  p = s 2 - 2s 4 - 8s 5 + 16s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 4 vs 6 5 vs 5

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y2, 0, y1, 3 y2 - 4 y1 - y4 + 3 y3, 2 y2 - 3 y1 + 2 y3, y4, 0, y3, 0]

 

  p' = s 3 - s 5   p = s 3 - s 5

Omega Rank for B :  cycles: {{2, 9}},   net cycles: 0 .    order:   4

[y1, y2, 0, y3, 0, 0, y4, 0, y5]  

See Matrices
 

 » SYNC'D 35/1024 , 0.03417968750

 
242 . Coloring, {3, 4, 5, 6, 7, 9}

R: [4, 4, 5, 8, 3, 8, 5, 1, 2]    B: [2, 9, 4, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 9` (` - 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [559, 172, 459, 663, 612, 153, 357, 612, 43] . FixedPtCheck, [559, 172, 459, 663, 612, 153, 357, 612, 43]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {1, 4, 8}},   net cycles: 1 .    order:   6

See Matrix
 

[-y1 + 2 y2 - y3 + 2 y4 - y5, y1, y2, y3, y4, 0, 0, y5, 0]

 

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y3, 0, 2 y3, y4, 0, y5]

 

  p = - s 3 + s 6

 » SYNC'D 1521/16384 , 0.09283447266

 
243 . Coloring, {3, 4, 5, 6, 8, 9}

R: [4, 4, 5, 8, 3, 8, 1, 6, 2]    B: [2, 9, 4, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` 5 + 3τ ` )`` (` 3 + τ 2 ` )` , -54` (` - 1 + τ ` )` 2 ` (` 5 + 3τ ` )` , 9` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 18` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 3 , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , 27` (` - 1 + τ ` )` 3 ` (` 5 + 3τ ` )``]`

For τ=1/2, [-338, -104, -102, -357, -136, -459, -238, -612, -26] . FixedPtCheck, [338, 104, 102, 357, 136, 459, 238, 612, 26]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 7 vs 8 4 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[3 y4, y4, y1, 2 y1 + 3 y2 - y3, y2, y3, 0, -4 y4 + 3 y1 + 2 y2, 0]

 

  p' = s 4 - s 6   p' = s 3 - s 5   p = s 3 - s 7

Omega Rank for B :  cycles: {{5, 7}, {1, 2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[4 y5, 4 y4, 0, 4 y3, 4 y2, 0, 5 y5 + 5 y4 - 4 y3 - 4 y2 + 5 y1, 0, 4 y1]

 

  p = s 2 + s 3 - s 5 - s 6

 » SYNC'D 5583/524288 , 0.01064872742

 
244 . Coloring, {3, 4, 5, 7, 8, 9}

R: [4, 4, 5, 8, 3, 7, 5, 6, 2]    B: [2, 9, 4, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , -18` (` - 1 + τ ` )` 3 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 3 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 4 ` (` 5 + 3τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [767, 236, 2295, 1326, 3060, 918, 1785, 1224, 59] . FixedPtCheck, [767, 236, 2295, 1326, 3060, 918, 1785, 1224, 59]

det(A + τ Δ) =   1` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )` 2 ` (` τ ` )` 2
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 7 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}},   net cycles: 0 .    order:   6

[0, y6, y1, y7, y2, y3, y4, y5, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y2, y1, 0, y4, 0, 0, y3, y4, y5]

 

  p = - s 3 + s 6

 » SYNC'D 63051/524288 , 0.1202602386

 
245 . Coloring, {3, 4, 6, 7, 8, 9}

R: [4, 4, 5, 8, 7, 8, 5, 6, 2]    B: [2, 9, 4, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-325, -100, -153, -357, -612, -459, -663, -612, -25] . FixedPtCheck, [325, 100, 153, 357, 612, 459, 663, 612, 25]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 6 vs 6

Omega Rank for R :  cycles: {{5, 7}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1, 0, 3 y1 - y4 - 4 y3 + 3 y2, 2 y1 - 3 y3 + 2 y2, y4, y3, y2, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   6

[y3, y4, y5, y6, 0, 0, y1, 0, y2]  

See Matrices
 

 » SYNC'D 61/1024 , 0.05957031250

 
246 . Coloring, {3, 5, 6, 7, 8, 9}

R: [4, 4, 5, 7, 3, 8, 5, 6, 2]    B: [2, 9, 4, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , -18` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 3 , 9` (` 1 + τ ` )` 4 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -18` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 4 `]`

For τ=1/2, [377, 116, 1377, 714, 1836, 306, 1071, 408, 29] . FixedPtCheck, [377, 116, 1377, 714, 1836, 306, 1071, 408, 29]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` 3
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, -y1 + y3 - y4 + 4 y5, y1, 4 y3 + y5 - y2, y2, y3, y4, y5, 0]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: -1 .    order:   3

See Matrix
 

[y1, y2, 0, y3, 0, 0, 3 y3, y5, y4]

 

  p = - s 3 + s 6

 » SYNC'D 6823/131072 , 0.05205535889

 
247 . Coloring, {4, 5, 6, 7, 8, 9}

R: [4, 4, 4, 8, 3, 8, 5, 6, 2]    B: [2, 9, 5, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [-1118, -344, -918, -1785, -1224, -2295, -1326, -3060, -86] . FixedPtCheck, [1118, 344, 918, 1785, 1224, 2295, 1326, 3060, 86]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 6 5 vs 5

Omega Rank for R :  cycles: {{6, 8}},   net cycles: -1 .    order:   4

See Matrix
 

[0, y1, y5, y4, 3 y1, y3, 0, y2, 0]

 

  p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   3

[y5, y4, 0, 0, y3, 0, y2, 0, y1]  

See Matrices
 

 » SYNC'D 915/8192 , 0.1116943359

 
248 . Coloring, {2, 3, 4, 5, 6, 7, 8}

R: [4, 9, 5, 8, 3, 8, 5, 6, 1]    B: [2, 4, 4, 7, 7, 7, 1, 1, 2]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , -18` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 18` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -9` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )``]`

For τ=1/2, [247, 76, 387, 301, 516, 387, 301, 516, 57] . FixedPtCheck, [247, 76, 387, 301, 516, 387, 301, 516, 57]

det(A + τ Δ) =   0

Delta Range :  [-y6 - y2 - y3 - y4 - y7, y6, y5, y2, y3, y4, y1, -y5 - y1, y7]

[3, 2, 1, 3, 2, 1, 3, 2, 1]

+              \ ;             -              \ ;             Δ

See Matrices

 
[y5 + y2 + y3 - 2 y4 - y1, -y5 - 2 y2 - 2 y3 + y4, -y5 - y6, y1, y2, y3, y5, y6, y4]
  p = s 3 - s 4 - 8s 7

            S+              \ ;             S-              \ ;             NM
See Matrices

CmmCk true, true, true


Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 7 7 vs 7 7 vs 7 5 vs 7 4 vs 4

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[y5, 0, y4, y3, 2 y5 - 3 y4 + 2 y1, y2, 0, y1, 3 y5 - 4 y4 - y3 - y2 + 3 y1]

 

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 1 .    order:   4

[y3, y1, 0, y2, 0, 0, y4, 0, 0]  

See Matrices
 

 » SYNC'D 615/32768 , 0.01876831055

 
249 . Coloring, {2, 3, 4, 5, 6, 7, 9}

R: [4, 9, 5, 8, 3, 8, 5, 1, 2]    B: [2, 4, 4, 7, 7, 7, 1, 6, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [301, 172, 225, 325, 300, 75, 175, 300, 129] . FixedPtCheck, [301, 172, 225, 325, 300, 75, 175, 300, 129]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 7 4 vs 5

Omega Rank for R :  cycles: {{1, 4, 8}, {3, 5}, {2, 9}},   net cycles: 3 .    order:   6

See Matrix
 

[3 y1 - y2 - y4 + 3 y3, y1, 2 y1, y2, 2 y3, 0, 0, y4, y3]

 

  p' = s 2 + s 3 - s 5 - s 6   p' = s - s 3 - s 4 + s 6   p = s - s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y3, y4, 0, y2, 0, y1, y3 - y4 + y2 + y1, 0, 0]

 

  p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 5859/262144 , 0.02235031128

 
250 . Coloring, {2, 3, 4, 5, 6, 8, 9}

R: [4, 9, 5, 8, 3, 8, 1, 6, 2]    B: [2, 4, 4, 7, 7, 7, 5, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `27` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ ` )` , 54` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 3τ ` )` , 9` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` 5 + 3τ ` )``]`

For τ=1/2, [-546, -312, -150, -525, -200, -675, -350, -900, -234] . FixedPtCheck, [546, 312, 150, 525, 200, 675, 350, 900, 234]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 8 4 vs 5

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[-8 y3 + 3 y4 + 3 y2 - y1, y3, %1, y4, y3, y2, 0, y1, %1] %1 := -3 y3 + y4 + y2

 

  p' = s 3 - s 7   p' = s 4 - s 6   p' = s 5 - s 7   p = s 3 - s 7

Omega Rank for B :  cycles: {{5, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 - y2 - y3 + y4, y1, 0, y2, y3, 0, y4, 0, 0]

 

  p = - s 4 + s 5

 » SYNC'D 3861/1048576 , 0.003682136536

 
251 . Coloring, {2, 3, 4, 5, 7, 8, 9}

R: [4, 9, 5, 8, 3, 7, 5, 6, 2]    B: [2, 4, 4, 7, 7, 8, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 18` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ 2 ` )`` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -18` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 + 3τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [413, 236, 1125, 650, 1500, 450, 875, 600, 177] . FixedPtCheck, [413, 236, 1125, 650, 1500, 450, 875, 600, 177]

det(A + τ Δ) =   1` (` τ ` )` 2 ` (` 1 + τ ` )` 4 ` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 6 vs 8 4 vs 5

Omega Rank for R :  cycles: {{3, 5}, {2, 9}},   net cycles: 1 .    order:   6

See Matrix
 

[0, y1, y2, -15 y1 - y3 - y4 + 4 y2 + 4 y5 + 4 y6, y3, y4, y5, y6, -4 y1 + y2 + y5 + y6]

 

  p' = s 5 - s 7   p = s 5 - s 7

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 - y2 + y3 + y4, y1, 0, y2, 0, 0, y3, y4, 0]

 

  p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 35637/1048576 , 0.03398609161

 
252 . Coloring, {2, 3, 4, 6, 7, 8, 9}

R: [4, 9, 5, 8, 7, 8, 5, 6, 2]    B: [2, 4, 4, 7, 3, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )` , 9` (` 3 + τ ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )` , -9` (` 1 + τ ` )` 2 , -9` (` 3 + τ 2 ` )` , -18` (` 1 + τ ` )` , 9` (` 1 + τ ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-7, -4, -3, -7, -12, -9, -13, -12, -3] . FixedPtCheck, [7, 4, 3, 7, 12, 9, 13, 12, 3]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 7 4 vs 5

Omega Rank for R :  cycles: {{2, 9}, {6, 8}, {5, 7}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y1 + y2 - 2 y3, 0, y1, 2 y3, y2, 2 y1 + 2 y2 - 4 y3, 2 y1 + 2 y2 - 3 y3, y3]

 

  p = s 2 - s 4   p' = - s 2 + s 4   p' = - s 3 + s 5   p' = - s 2 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y1 + y2 - y4 + y3, y1, y2, y4, 0, 0, y3, 0, 0]

 

  p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 801/131072 , 0.006111145020

 
253 . Coloring, {2, 3, 5, 6, 7, 8, 9}

R: [4, 9, 5, 7, 3, 8, 5, 6, 2]    B: [2, 4, 4, 8, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 + 4τ + τ 2 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 , 18` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )`` (` 5 + 4τ + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [203, 116, 675, 350, 900, 150, 525, 200, 87] . FixedPtCheck, [203, 116, 675, 350, 900, 150, 525, 200, 87]

det(A + τ Δ) =   1` (` 1 + τ ` )` 4 ` (` τ ` )` 2 ` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 9 vs 9 9 vs 9 4 vs 8 4 vs 5

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   4

See Matrix
 

[0, y3, 3 y3 + y4 - y2, y3 - y1 + 3 y4, y1, y4, y2, y3, y4]

 

  p' = - s 3 + s 7   p' = - s 3 + s 5   p = - s 3 + s 7   p = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 4, 8}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, y1, 0, -y2 + y1 + y4 + y3, 0, 0, y4, y3, 0]

 

  p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 675/131072 , 0.005149841309

 
254 . Coloring, {2, 4, 5, 6, 7, 8, 9}

R: [4, 9, 4, 8, 3, 8, 5, 6, 2]    B: [2, 4, 5, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 18` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 9` (` 1 + τ 2 ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -9` (` 1 + τ ` )` 2 ` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -18` (` 1 + τ ` )`` (` 1 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-602, -344, -450, -875, -600, -1125, -650, -1500, -258] . FixedPtCheck, [602, 344, 450, 875, 600, 1125, 650, 1500, 258]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 7 vs 7 7 vs 7 5 vs 7 4 vs 5

Omega Rank for R :  cycles: {{2, 9}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y4, y3, y2, y1, y4 - y2 - y1 + 4 y5, 0, 4 y4 - y3 + y5, y5]

 

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 0 .    order:   4

See Matrix
 

[y2, y3, 0, y4, -y2 + y3 - y4 + y1, 0, y1, 0, 0]

 

  p = - s 2 + s 3 - s 4 + s 5

 » SYNC'D 485/16384 , 0.02960205078

 
255 . Coloring, {3, 4, 5, 6, 7, 8, 9}

R: [4, 4, 5, 8, 3, 8, 5, 6, 2]    B: [2, 9, 4, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -18` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -9` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 18` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 9` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 3 `]`

For τ=1/2, [247, 76, 459, 357, 612, 459, 357, 612, 19] . FixedPtCheck, [247, 76, 459, 357, 612, 459, 357, 612, 19]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 4 vs 6 5 vs 5

Omega Rank for R :  cycles: {{3, 5}, {6, 8}},   net cycles: 1 .    order:   4

See Matrix
 

[0, y1, y2, 3 y1 - 4 y2 - y3 + 3 y4, 2 y1 - 3 y2 + 2 y4, y3, 0, y4, 0]

 

  p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{1, 2, 9}},   net cycles: 0 .    order:   3

[y1, y4, 0, y5, 0, 0, y3, 0, y2]  

See Matrices
 

 » SYNC'D 215/4096 , 0.05249023438

 
256 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9}

R: [4, 9, 5, 8, 3, 8, 5, 6, 2]    B: [2, 4, 4, 7, 7, 7, 1, 1, 1]

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 18` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , -9` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 18` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 9` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [133, 76, 225, 175, 300, 225, 175, 300, 57] . FixedPtCheck, [133, 76, 225, 175, 300, 225, 175, 300, 57]

det(A + τ Δ) =   0
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 7 8 vs 8 8 vs 8 3 vs 7 3 vs 4

Omega Rank for R :  cycles: {{3, 5}, {2, 9}, {6, 8}},   net cycles: 2 .    order:   2

See Matrix
 

[0, y2, 2 y2, y1, 2 y3, y2 - y1 + 2 y3, 0, 2 y2 + y3, y3]

 

  p' = - s 2 + s 6   p = - s 2 + s 6   p' = - s 2 + s 4   p = - s 2 + s 4

Omega Rank for B :  cycles: {{1, 2, 4, 7}},   net cycles: 1 .    order:   4

See Matrix
 

[y2 - y1 + y3, y2, 0, y1, 0, 0, y3, 0, 0]

 

  p = - s + s 2 - s 3 + s 4

 » SYNC'D 405/65536 , 0.006179809570

 
SUMMARY
Graph Type
CC
ν(A)
 2
ν(Δ)
 2
π
 [3, 2, 1, 3, 2, 1, 3, 2, 1]
Dbly Stoch
false

 
SANDWICH
Total 4
No .ColoringRank
1 {2, 5, 8} 3
2 {} 3
3 {3, 6, 9} 3
4 {2, 3, 5, 6, 8, 9} 6

 
RT GROUPS
Total 1
No .ColoringRankSolv
1 {2, 4, 7, 9} 2 Not Solvable

 
CC Colorings
Total 1
No .ColoringSandwich,Rank
1 {} true, 3

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
213 0 243 , 247 28 , 44 5 256 256