New Graph

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

 

 

π = [6, 3, 5, 6, 4, 3]

POSSIBLE RANKS

1 x 27
3 x 9

BASE DETERMINANT 161757/1048576, .1542634964

NullSpace of Δ

{1, 2, 3, 4, 5, 6}

Nullspace of A

` det(A) = ` 1/32

 

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-18` (` - 3 + τ ` )` , 9` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 9` (` 5 - 2τ + τ 2 ` )` , -18` (` - 3 + τ ` )` , -36` (` - 1 + τ ` )` , 9` (` - 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [20, 5, 17, 20, 8, 5] . FixedPtCheck, [20, 5, 17, 20, 8, 5]

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

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

[6, 3, 5, 6, 4, 3]

+              \ ;      -              \ ;      Δ

See Matrices

 
[-y2, y2, -y1, -y2, y1, y2]
  p' = s 2 - 4s 3 + 4s 4   p = s - 32s 4 + 48s 5

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true

  p' = s - 12s 3 + 16s 4
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
2 vs 5 3 vs 6 3 vs 6 2 vs 4 2 vs 6

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

See Matrix
 

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

 

  p = - s 2 + s 3   p = - s 2 + s 4

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

See Matrix
 

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

 

  p' = - s + s 5   p' = - 1 + s 4   p' = - s + s 3   p' = - 1 + s 2


  « NOT SYNC'D »

Nullspace of {Ω&Deltai} :
[x1, x2, x3, -32 x1 - 12 x2 - 4 x3, 48 x1 + 16 x2 + 4 x3]
For A+2Δ :   [-y1 - y2, -y2 - y3, y2, y1, y2, y3]
For A-2Δ :   [-y1 - y2, -y2 - y3, y2, y1, y2, y3]

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

 
rank of M is 6 , rank of N is 4

M               N

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

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

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

τ= 12 , r'= 2/3

Ranges

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

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

Partitions

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

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

b1 = {4, 6} ` , ` b2 = {2, 4} ` , ` b3 = {1, 2} ` , ` b4 = {3, 5} ` , ` b5 = {1, 6}

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

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

` ` See level-3 partition graph.

`

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

 

 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` - 15 - 11τ - 14τ 2 + 10τ 3 - 3τ 4 + τ 5 ` )` , -54` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` 2 , -108` (` - 1 + τ ` )`` (` - 3 - τ - τ 2 + τ 3 ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` 2 `]`

For τ=1/2, [-780, -195, -733, -900, -232, -225] . FixedPtCheck, [780, 195, 733, 900, 232, 225]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y3, 0, y2, y1, 0, 0]  

See Matrices
 

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

[y1, y4, y2, 0, y3, y5]  

See Matrices
 

 » SYNC'D 13/64 , 0.2031250000

 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -27` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -54` (` 3 + τ 2 ` )` , -108` (` 1 + τ ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-52, -13, -51, -52, -48, -13] . FixedPtCheck, [52, 13, 51, 52, 48, 13]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y3, y2, y4, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 4, 6}},   net cycles: 0 .    order:   4

[y3, y4, y5, y1, 0, y2]  

See Matrices
 

 » SYNC'D 321/2048 , 0.1567382812

 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 3 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 3 + τ ` )` , 27` (` 15 - 2τ + 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )` 2 ` (` 3 + 2τ + τ 2 ` )` , 27` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-420, -105, -229, -780, -136, -585] . FixedPtCheck, [420, 105, 229, 780, 136, 585]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, 0, 0, y3, y1, y2]  

See Matrices
 

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

[y1, y5, y4, y3, y2, 0]  

See Matrices
 

 » SYNC'D 9/128 , 0.07031250000

 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 27` (` 1 + τ ` )`` (` - 5 + τ 2 ` )` , -54` (` 3 + τ 2 ` )` , 108` (` 1 + τ ` )`` (` - 1 + τ ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-52, -13, -57, -52, -24, -13] . FixedPtCheck, [52, 13, 57, 52, 24, 13]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, 0, y3, y2, y1, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 4, 6}},   net cycles: 0 .    order:   4

[y1, y5, 0, y4, y3, y2]  

See Matrices
 

 » SYNC'D 99/512 , 0.1933593750

 
6 . Coloring, {6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 1 + τ ` )`` (` - 3 + τ ` )` 2 , 27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` 2 , 27` (` - 15 + 4τ - 2τ 2 - 4τ 3 + τ 4 ` )` , 54` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 108` (` - 1 + τ ` )`` (` 3 + τ ` )` , -27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-300, -75, -223, -260, -112, -65] . FixedPtCheck, [300, 75, 223, 260, 112, 65]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, 0, y3, y2, y1, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s 2 + s 4   p' = - s 2 + s 4

 » SYNC'D 21/512 , 0.04101562500

 
7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` 2 , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` 2 , 27` (` - 15 - 11τ - 14τ 2 + 10τ 3 - 3τ 4 + τ 5 ` )` , 54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 108` (` - 3 - τ - 5τ 2 + τ 3 ` )` , -27` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-676, -169, -733, -780, -592, -195] . FixedPtCheck, [676, 169, 733, 780, 592, 195]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, 0, y3, y2, y1, 0]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 3, 5}},   net cycles: 0 .    order:   4

[y3, y1, y2, 0, y4, y5]  

See Matrices
 

 » SYNC'D 219/1024 , 0.2138671875

 
8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 3 + τ ` )` , -27` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )`` (` - 3 + τ ` )` , 27` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 54` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` 2 , -108` (` - 1 + τ ` )` 2 ` (` - 3 + τ 2 ` )` , 27` (` 1 + τ ` )` 3 ` (` - 3 + τ ` )` 2 `]`

For τ=1/2, [420, 105, 247, 900, 88, 675] . FixedPtCheck, [420, 105, 247, 900, 88, 675]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{4, 6}},   net cycles: 0 .    order:   2

[y1, 0, 0, y2, 0, y3]  

See Matrices
 

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

[y2, y1, y4, 0, y3, 0]  

See Matrices
 

 » SYNC'D 3/32 , 0.09375000000

 
9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 3 + τ 2 ` )` 2 , -27` (` 3 + τ 2 ` )` 2 ` (` - 1 + τ ` )` , 27` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -54` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 108` (` - 3 - τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 27` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [676, 169, 759, 780, 232, 195] . FixedPtCheck, [676, 169, 759, 780, 232, 195]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y2, y3, 0, 0]  

See Matrices
 

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

[y1, y4, 0, 0, y3, y2]  

See Matrices
 

 » SYNC'D 41/128 , 0.3203125000

 
10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , -27` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -54` (` 1 + τ ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [60, 15, 49, 60, 16, 15] . FixedPtCheck, [60, 15, 49, 60, 16, 15]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y3, 0, y1, y2, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s 2 + s 4   p' = - s 2 + s 4

 » SYNC'D 27/512 , 0.05273437500

 
11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 3 + τ 2 ` )` , -27` (` 15 - 2τ + 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 54` (` 3 + τ 2 ` )` 2 , 108` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` 3 + τ 2 ` )` 2 ` (` 1 + τ ` )``]`

For τ=1/2, [364, 91, 229, 676, 240, 507] . FixedPtCheck, [364, 91, 229, 676, 240, 507]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y2, 0, 0, y1, y4, y3]  

See Matrices
 

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

[y1, y2, y4, y3, 0, 0]  

See Matrices
 

 » SYNC'D 231/1024 , 0.2255859375

 
12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` - 1 + τ ` )`` (` 3 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` , -27` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , -54` (` - 1 + τ ` )`` (` 3 + τ ` )` , 108` (` 1 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )``]`

For τ=1/2, [28, 7, 57, 28, 48, 7] . FixedPtCheck, [28, 7, 57, 28, 48, 7]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, 0, y3, y2, y1, 0]  

See Matrices
 

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

[y1, y2, 0, y3, 0, y4]  

See Matrices
 

 » SYNC'D 75/256 , 0.2929687500

 
13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , -27` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 27` (` - 15 + 4τ - 2τ 2 - 4τ 3 + τ 4 ` )`` (` 1 + τ ` )` , -54` (` 3 + τ 2 ` )` 2 , -108` (` 1 + τ ` )` 2 ` (` 3 - 2τ + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` 2 `]`

For τ=1/2, [-780, -195, -669, -676, -648, -169] . FixedPtCheck, [780, 195, 669, 676, 648, 169]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y3, y2, y4, 0]  

See Matrices
 

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

[y1, y2, y3, y5, 0, y4]  

See Matrices
 

 » SYNC'D 343/4096 , 0.08374023438

 
14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 27` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 27` (` - 15 - 4τ - 2τ 2 + 4τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 54` (` 3 + τ 2 ` )` 2 , 108` (` 3 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` 2 , 27` (` 1 + τ ` )`` (` 3 + τ 2 ` )` 2 `]`

For τ=1/2, [364, 91, 271, 676, 136, 507] . FixedPtCheck, [364, 91, 271, 676, 136, 507]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y2, y3, y4, y5]  

See Matrices
 

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

[y5, y4, y1, y2, y3, 0]  

See Matrices
 

 » SYNC'D 45/1024 , 0.04394531250

 
15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -54` (` 1 + τ ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )` , -27` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )``]`

For τ=1/2, [60, 15, 19, 60, 16, 45] . FixedPtCheck, [60, 15, 19, 60, 16, 45]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y4, 0, 0, y3, y1, y2]  

See Matrices
 

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

[0, y4, y3, y2, y1, 0]  

See Matrices
 

 » SYNC'D 39/256 , 0.1523437500

 
16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 27` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` 1 + τ ` )`` (` 15 + 2τ - 2τ 3 + τ 4 ` )` , 54` (` 3 + τ 2 ` )` 2 , -108` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , -27` (` 3 + τ 2 ` )` 2 ` (` - 1 + τ ` )``]`

For τ=1/2, [780, 195, 759, 676, 336, 169] . FixedPtCheck, [780, 195, 759, 676, 336, 169]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y3, 0, y4, y2, y1, 0]  

See Matrices
 

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

[y1, y4, 0, y2, y3, y5]  

See Matrices
 

 » SYNC'D 221/2048 , 0.1079101562

 
17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )`` (` - 1 + τ ` )` , 27` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 , 27` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , -54` (` 1 + τ ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )` , -27` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )``]`

For τ=1/2, [28, 7, 19, 60, 16, 45] . FixedPtCheck, [28, 7, 19, 60, 16, 45]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y3, 0, 0, y2, y1, y4]  

See Matrices
 

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

[y2, y3, y4, 0, y1, 0]  

See Matrices
 

 » SYNC'D 241/1024 , 0.2353515625

 
18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 27` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` 2 , 27` (` 1 + τ ` )`` (` 15 + 2τ - 2τ 3 + τ 4 ` )` , 54` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -108` (` - 3 - τ - 5τ 2 + τ 3 ` )` , -27` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 3 + τ ` )``]`

For τ=1/2, [364, 91, 759, 420, 592, 105] . FixedPtCheck, [364, 91, 759, 420, 592, 105]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[0, 0, y3, y1, y2, 0]  

See Matrices
 

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

[y3, y4, 0, 0, y2, y1]  

See Matrices
 

 » SYNC'D 25/64 , 0.3906250000

 
19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 27` (` - 5 - τ - 3τ 2 + τ 3 ` )` , -54` (` 3 + τ 2 ` )` , -108` (` 1 + τ 2 ` )` , 27` (` 3 + τ 2 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-52, -13, -49, -52, -40, -13] . FixedPtCheck, [52, 13, 49, 52, 40, 13]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y2, y4, y3, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p' = - s + s 5   p' = - 1 + s 4

 » SYNC'D 479/16384 , 0.02923583984

 
20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 3 + τ 2 ` )` , -27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 15 - 9τ + τ 2 + τ 3 ` )` , 54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 108` (` - 1 + τ ` )` 2 ` (` - 3 + τ 2 ` )` , 27` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-364, -91, -261, -780, -88, -585] . FixedPtCheck, [364, 91, 261, 780, 88, 585]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y4, 0, y3, y2, 0, y1]  

See Matrices
 

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

[y1, y3, y4, 0, y2, 0]  

See Matrices
 

 » SYNC'D 31/256 , 0.1210937500

 
21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , 27` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , -27` (` 15 + 2τ - 2τ 3 + τ 4 ` )`` (` - 1 + τ ` )` , 54` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )` 2 , 108` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 , 27` (` 1 + τ ` )` 3 ` (` - 3 + τ ` )` 2 `]`

For τ=1/2, [780, 195, 253, 900, 112, 675] . FixedPtCheck, [780, 195, 253, 900, 112, 675]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, 0, y2, 0, y3]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s 2 + s 4

 » SYNC'D 27/128 , 0.2109375000

 
22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , -27` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -54` (` 3 + τ 2 ` )` , 108` (` - 1 + τ ` )` , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )``]`

For τ=1/2, [-52, -13, -51, -52, -16, -13] . FixedPtCheck, [52, 13, 51, 52, 16, 13]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y3, y2, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 205/4096 , 0.05004882812

 
23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` - 1 + τ ` )`` (` 3 + τ ` )` 2 , 27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` 2 , -27` (` - 15 - 4τ - 2τ 2 + 4τ 3 + τ 4 ` )` , 54` (` 3 + τ ` )`` (` 3 + τ 2 ` )` , -108` (` 1 + τ ` )`` (` - 3 + τ ` )` , 27` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 3 + τ 2 ` )``]`

For τ=1/2, [196, 49, 271, 364, 240, 273] . FixedPtCheck, [196, 49, 271, 364, 240, 273]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

See Matrix
 

[0, 0, 16 y2 - 16 y1, 4 y2, 4 y1, 16 y2 - 15 y1]

 

  p = - s + s 3   p' = - s + s 3

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

[y1, y2, y4, y3, 0, 0]  

See Matrices
 

 » SYNC'D 81/2048 , 0.03955078125

 
24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 3 + τ 2 ` )` , -27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` , 54` (` 3 + τ 2 ` )` , -108` (` - 1 + τ ` )`` (` 1 + τ ` )` , 27` (` 3 + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [52, 13, 19, 52, 24, 39] . FixedPtCheck, [52, 13, 19, 52, 24, 39]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y1, 0, 0, y2, y3, y4]  

See Matrices
 

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

[y1, y3, y2, y4, 0, 0]  

See Matrices
 

 » SYNC'D 171/512 , 0.3339843750

 
25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -27` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 ` (` - 3 + τ ` )` , 27` (` 1 + τ ` )`` (` 15 + 2τ - 2τ 3 + τ 4 ` )` , -54` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 108` (` 1 + τ ` )` 2 ` (` 3 - 2τ + τ 2 ` )` , 27` (` 3 + τ ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` 2 `]`

For τ=1/2, [420, 105, 759, 364, 648, 91] . FixedPtCheck, [420, 105, 759, 364, 648, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y3, y4, y2, 0]  

See Matrices
 

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

[y1, y2, 0, y3, 0, y4]  

See Matrices
 

 » SYNC'D 5/32 , 0.1562500000

 
26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -54` (` 3 + τ 2 ` )` , 108` (` - 1 + τ ` )` , -27` (` 3 + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-52, -13, -25, -52, -16, -39] . FixedPtCheck, [52, 13, 25, 52, 16, 39]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y2, 0, y1, y3, y4, y5]  

See Matrices
 

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

[y1, y2, y3, y4, y5, 0]  

See Matrices
 

 » SYNC'D 447/8192 , 0.05456542969

 
27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )` 2 ` (` - 1 + τ ` )` , 27` (` 3 + τ ` )` 2 ` (` - 1 + τ ` )` 2 , 27` (` 15 - 9τ + τ 2 + τ 3 ` )`` (` 1 + τ ` )` , -54` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 108` (` 3 + τ 2 ` )` , -27` (` 3 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )``]`

For τ=1/2, [196, 49, 261, 420, 208, 315] . FixedPtCheck, [196, 49, 261, 420, 208, 315]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

See Matrix
 

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

 

  p = s - s 3   p' = - s + s 3

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

[y3, y4, y2, 0, y1, 0]  

See Matrices
 

 » SYNC'D 33/512 , 0.06445312500

 
28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` 2 , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` 2 , 27` (` - 1 + τ ` )`` (` 15 + 2τ - 2τ 3 + τ 4 ` )` , 54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )`` (` - 3 - τ - τ 2 + τ 3 ` )` , 27` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-676, -169, -253, -780, -232, -585] . FixedPtCheck, [676, 169, 253, 780, 232, 585]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y1, 0, 0, y2, y3, y4]  

See Matrices
 

Omega Rank for B :  cycles: {{1, 2, 3, 5}},   net cycles: 0 .    order:   4

[y1, y4, y3, y2, y5, 0]  

See Matrices
 

 » SYNC'D 381/2048 , 0.1860351562

 
29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` - 1 + τ ` )`` (` 3 + τ ` )` , 27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` , 27` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -54` (` - 1 + τ ` )`` (` 3 + τ ` )` , 108` (` 1 + τ 2 ` )` , 27` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )``]`

For τ=1/2, [28, 7, 51, 28, 40, 7] . FixedPtCheck, [28, 7, 51, 28, 40, 7]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

[y1, 0, y4, y3, y2, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p = s + s 2 - s 4 - s 5

 » SYNC'D 87/1024 , 0.08496093750

 
30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ 2 ` )` 2 , 27` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` 2 , -27` (` - 15 + 7τ - τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` , 54` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` - 3 + τ ` )` , -108` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` , 27` (` 1 + τ ` )` 2 ` (` 3 + τ 2 ` )`` (` - 3 + τ ` )``]`

For τ=1/2, [-676, -169, -279, -780, -112, -585] . FixedPtCheck, [676, 169, 279, 780, 112, 585]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 4, 6}},   net cycles: 0 .    order:   3

[y1, 0, y4, y3, 0, y2]  

See Matrices
 

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

[y1, y2, y3, y4, y5, 0]  

See Matrices
 

 » SYNC'D 435/4096 , 0.1062011719

 
31 . Coloring, {3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `54` (` 3 + τ ` )` , -27` (` 3 + τ ` )`` (` - 1 + τ ` )` , 27` (` 5 + 2τ + τ 2 ` )` , 54` (` 3 + τ ` )` , 108` (` 1 + τ ` )` , 27` (` 3 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [28, 7, 25, 28, 24, 21] . FixedPtCheck, [28, 7, 25, 28, 24, 21]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

See Matrix
 

[5 y4 - 4 y3 + 5 y2 - 4 y1, 0, 4 y4, 4 y3, 4 y2, 4 y1]

 

  p = - s - s 2 + s 4 + s 5

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

[y3, y4, y1, y2, 0, 0]  

See Matrices
 

 » SYNC'D 81/2048 , 0.03955078125

 
32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-54` (` 3 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 27` (` - 15 + 7τ - τ 2 + τ 3 ` )`` (` 1 + τ ` )` , 54` (` 3 + τ ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 108` (` - 3 - τ - τ 2 + τ 3 ` )` , 27` (` 3 + τ ` )`` (` 1 + τ ` )` 2 ` (` - 3 + τ ` )``]`

For τ=1/2, [-364, -91, -279, -420, -232, -315] . FixedPtCheck, [364, 91, 279, 420, 232, 315]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1, 1]

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

See Matrix
 

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

 

  p = s + s 2 - s 4 - s 5

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

See Matrix
 

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

 

  p' = - s 3 + s 4   p = s 3 - s 4

 » SYNC'D 387/4096 , 0.09448242188

 
SUMMARY
Graph Type
NOT CC
ν(A)
 0
ν(Δ)
 1
π
 [6, 3, 5, 6, 4, 3]
Dbly Stoch
false

 
SANDWICH
Total 1
No .ColoringRank
1 {} 3

 
RT GROUPS
Total 0
No .ColoringRankSolv

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
31 0 31 , 31 27 , 24 1 32 32