New Graph

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

 

 

π = [3, 3, 3, 1, 3, 2]

POSSIBLE RANKS

1 x 15
3 x 5

BASE DETERMINANT 163959/1048576, .1563634872

NullSpace of Δ

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

Nullspace of A

` det(A) = ` 1/32

 

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-5, -5, -5, -1, -5, -4] . FixedPtCheck, [5, 5, 5, 1, 5, 4]

det(A + τ Δ) =   1` (` 1 + 10τ 2 + 5τ 4 ` )`` (` - 1 + τ ` )`

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

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

+              \ ;      -              \ ;      Δ

See Matrices

 
[0, 0, 0, y1, 0, -y1]
  p = s - 16s 5

         S+              \ ;      S-              \ ;      NM
See Matrices

CmmCk true, true, true

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

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

See Matrix
 

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

 

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

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

See Matrix
 

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

 

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


  « NOT SYNC'D »

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

Range of {ΩΔi}: [0, 0, 0, -μ1, 0, μ1]

 
rank of M is 5 , rank of N is 5

M               N

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

Check is ΩΔN zero? true, πΔ= [0, 0, 0, -1, 0, 1]

ker M, [0, 0, 0, -2 λ1, 0, λ1]
Range M, [x1, x2, x5, x3, x4, 2 x3]

τ= 8 , r'= 4/5

Ranges

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

ker N, [0, 0, 0, μ1, 0, -μ1]
Range of N
    [y5, y4, y2, y3, y1, y3]

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

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

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

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

` ` See level-5 partition graph.

`

Right Group
Coloring {}
Rank5
R,B [3, 5, 6, 1, 2, 1], [5, 1, 2, 6, 3, 4]
π2 [3, 3, 1, 3, 2, 3, 1, 3, 2, 1, 3, 2, 1, 0, 2]
u2 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1] (dim 1)
wpp [1, 1, 1, 2, 1, 2]
π5 [1, 0, 2, 0, 0, 0]
u5 [1, 0, 1, 0, 0, 0]

 

 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -15` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 15` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -30` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [-255, -125, -215, -43, -95, -172] . FixedPtCheck, [255, 125, 215, 43, 95, 172]

det(A + τ Δ) =   1` (` 1 + τ ` )`` (` 1 + 3τ 2 ` )`` (` - 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 5

See Matrix for A+τΔ

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

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

[y4, y3, y2, 0, 0, y1]  

See Matrices
 

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

See Matrix
 

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

 

  p = s + s 2 - s 4 - s 5

 » SYNC'D 783/4096 , 0.1911621094

 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -15` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 15` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -30` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [-57, -153, -75, -5, -129, -20] . FixedPtCheck, [57, 153, 75, 5, 129, 20]

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: {{2, 5}},   net cycles: 0 .    order:   4

[y1, y4, y2, 0, y3, 0]  

See Matrices
 

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

[y5, 0, y4, y1, y2, y3]  

See Matrices
 

 » SYNC'D 25/256 , 0.09765625000

 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 1 + τ ` )` , 15` (` 3 + τ 2 ` )` , 30` (` 1 + τ ` )``]`

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

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

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p = - s + s 4   p' = - s + s 4

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

[y2, y1, y3, y5, y4, 0]  

See Matrices
 

 » SYNC'D 85/4096 , 0.02075195312

 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 15` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -30` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-215, -95, -255, -51, -125, -204] . FixedPtCheck, [215, 95, 255, 51, 125, 204]

det(A + τ Δ) =   1` (` - 1 + τ ` )` 2 ` (` 1 + 3τ 2 ` )`` (` 1 + τ ` )`
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: {{1, 3, 6}},   net cycles: 0 .    order:   3

[y4, 0, y3, 0, y2, y1]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

 » SYNC'D 819/4096 , 0.1999511719

 
6 . Coloring, {6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )` 2 , 15` (` 3 + τ 2 ` )` , 30` (` 1 + τ ` )``]`

For τ=1/2, [13, 13, 13, 9, 13, 12] . FixedPtCheck, [13, 13, 13, 9, 13, 12]

det(A + τ Δ) =   1` (` 1 + 10τ 2 + 5τ 4 ` )`` (` 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 4 vs 6 5 vs 5

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p' = - 1 + s 4   p' = - s + s 5

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

[y5, y1, y2, 0, y3, y4]  

See Matrices
 

 » SYNC'D 15/512 , 0.02929687500

 
7 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -15` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )`` (` - 3 + τ ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , 15` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , -30` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [735, 765, 645, 43, 375, 172] . FixedPtCheck, [735, 765, 645, 43, 375, 172]

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, 2, 3}},   net cycles: 1 .    order:   3

[y2, y3, y1, 0, 0, 0]  

See Matrices
 

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

[0, 0, y1, y2, y3, y4]  

See Matrices
 

 » SYNC'D 19/64 , 0.2968750000

 
8 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -15` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 15` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 30` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [663, 325, 559, 129, 247, 516] . FixedPtCheck, [663, 325, 559, 129, 247, 516]

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, 6}},   net cycles: 0 .    order:   3

[y3, y2, y1, 0, 0, y4]  

See Matrices
 

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

[y2, y1, y3, y4, y5, 0]  

See Matrices
 

 » SYNC'D 245/1024 , 0.2392578125

 
9 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` - 3 + τ ` )` , -15` (` - 1 + τ ` )`` (` - 3 + τ ` )` , 15` (` 1 + τ ` )`` (` - 3 + τ ` )` , 15` (` 1 + τ ` )`` (` - 1 + τ ` )` , -15` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -30` (` 1 + τ ` )``]`

For τ=1/2, [-15, -5, -15, -3, -5, -12] . FixedPtCheck, [15, 5, 15, 3, 5, 12]

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 2 vs 4

See Matrix for A+τΔ

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

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

[y1, 0, y3, 0, 0, y2]  

See Matrices
 

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

See Matrix
 

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

 

  p = - s + s 3   p' = - s + s 3

 » SYNC'D 15/512 , 0.02929687500

 
10 . Coloring, {2, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )` 2 ` (` 5 - τ + 3τ 2 + τ 3 ` )` , 15` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 3 + τ 2 ` )` , 30` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [663, 325, 559, 387, 247, 516] . FixedPtCheck, [663, 325, 559, 387, 247, 516]

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 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, 3, 4, 6}},   net cycles: 0 .    order:   4

[y1, y2, y3, y4, 0, y5]  

See Matrices
 

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

[y3, y4, y1, 0, y2, y5]  

See Matrices
 

 » SYNC'D 357/4096 , 0.08715820312

 
11 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )` 3 ` (` 5 + 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 30` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [247, 663, 325, 25, 559, 100] . FixedPtCheck, [247, 663, 325, 25, 559, 100]

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: {{2, 5}},   net cycles: 0 .    order:   4

[y4, y5, y2, 0, y3, y1]  

See Matrices
 

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

[y1, 0, y2, y3, y4, y5]  

See Matrices
 

 » SYNC'D 515/8192 , 0.06286621094

 
12 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -15` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` - 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , -15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , -30` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [125, 245, 255, 17, 215, 68] . FixedPtCheck, [125, 245, 255, 17, 215, 68]

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 2 vs 5

See Matrix for A+τΔ

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

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

[y3, y2, y1, 0, y4, 0]  

See Matrices
 

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

See Matrix
 

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

 

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

 » SYNC'D 135/2048 , 0.06591796875

 
13 . Coloring, {3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , 15` (` 1 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 30` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [247, 663, 325, 75, 559, 100] . FixedPtCheck, [247, 663, 325, 75, 559, 100]

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

See Matrix for A+τΔ

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

Omega Rank for R :  cycles: {{2, 5}},   net cycles: 0 .    order:   4

[y1, y2, y4, y5, y3, 0]  

See Matrices
 

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

[y1, 0, y2, 0, y3, y4]  

See Matrices
 

 » SYNC'D 111/1024 , 0.1083984375

 
14 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 30` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [559, 247, 663, 153, 325, 612] . FixedPtCheck, [559, 247, 663, 153, 325, 612]

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, 3, 6}},   net cycles: 0 .    order:   3

[y1, 0, y4, 0, y3, y2]  

See Matrices
 

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

[y4, y3, 0, y1, y2, 0]  

See Matrices
 

 » SYNC'D 179/512 , 0.3496093750

 
15 . Coloring, {4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ ` )`` (` - 1 + τ ` )` , 15` (` 3 + τ ` )`` (` - 1 + τ ` )` , 15` (` 3 + τ ` )`` (` - 1 + τ ` )` , -15` (` 1 + τ ` )` 2 , 15` (` 3 + τ ` )`` (` - 1 + τ ` )` , -30` (` 1 + τ ` )``]`

For τ=1/2, [-7, -7, -7, -9, -7, -12] . FixedPtCheck, [7, 7, 7, 9, 7, 12]

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

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

See Matrix
 

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

 

  p = s 2 - s 4   p' = - s 2 + s 4

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

[y1, y2, y3, 0, y4, 0]  

See Matrices
 

 » SYNC'D 75/2048 , 0.03662109375

 
16 . Coloring, {5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ 2 ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 15` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -15` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , 30` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [559, 247, 663, 459, 325, 612] . FixedPtCheck, [559, 247, 663, 459, 325, 612]

det(A + τ Δ) =   1` (` 1 + τ ` )` 2 ` (` 1 + 3τ 2 ` )`` (` - 1 + τ ` )`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 5 6 vs 6 6 vs 6 5 vs 5 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, 6}},   net cycles: 0 .    order:   4

[y1, 0, y2, y4, y3, y5]  

See Matrices
 

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

[y1, y2, 0, 0, y3, y4]  

See Matrices
 

 » SYNC'D 645/2048 , 0.3149414062

 
17 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` 2 , -15` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 1 + τ ` )` , -30` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [637, 663, 559, 43, 325, 172] . FixedPtCheck, [637, 663, 559, 43, 325, 172]

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, 2, 3}},   net cycles: 0 .    order:   3

[y3, y2, y1, 0, 0, y4]  

See Matrices
 

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

[y1, 0, y2, y3, y4, y5]  

See Matrices
 

 » SYNC'D 1215/4096 , 0.2966308594

 
18 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` - 3 + τ ` )` , 15` (` 1 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 3 + τ ` )` , -15` (` - 1 + τ ` )` 2 ` (` - 5 + τ 2 ` )` , 15` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )`` (` - 3 + τ ` )` , 30` (` - 1 + τ ` )`` (` - 5 + τ 2 ` )``]`

For τ=1/2, [255, 245, 285, 19, 125, 76] . FixedPtCheck, [255, 245, 285, 19, 125, 76]

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 2 vs 4

See Matrix for A+τΔ

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

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

[y1, y2, y3, 0, 0, 0]  

See Matrices
 

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

See Matrix
 

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

 

  p' = - s + s 3   p = - s + s 3

 » SYNC'D 27/512 , 0.05273437500

 
19 . Coloring, {2, 3, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -15` (` 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , 15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 30` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-637, -663, -559, -129, -325, -172] . FixedPtCheck, [637, 663, 559, 129, 325, 172]

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, 2, 3}},   net cycles: 0 .    order:   3

[y4, y1, y2, y3, 0, 0]  

See Matrices
 

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

[y2, 0, y1, 0, y3, y4]  

See Matrices
 

 » SYNC'D 135/512 , 0.2636718750

 
20 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -15` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )` 2 ` (` - 1 + τ ` )` , 15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , -30` (` 1 + τ ` )` 2 `]`

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

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, 6}},   net cycles: 1 .    order:   3

[y1, 0, y3, 0, 0, y2]  

See Matrices
 

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

[y1, y2, 0, y3, y4, 0]  

See Matrices
 

 » SYNC'D 7/128 , 0.05468750000

 
21 . Coloring, {2, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -15` (` 3 + τ ` )`` (` - 1 + τ ` )` 2 ` (` 5 + 2τ + τ 2 ` )` , 15` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )` , -15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )` 2 , 15` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 , -30` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-357, -175, -301, -387, -133, -516] . FixedPtCheck, [357, 175, 301, 387, 133, 516]

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 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, y3, y1, y5, 0, y4]  

See Matrices
 

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

[y1, y2, y3, 0, y4, 0]  

See Matrices
 

 » SYNC'D 77/512 , 0.1503906250

 
22 . Coloring, {2, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , -15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 15` (` 1 + τ ` )`` (` 3 + τ 2 ` )` , 15` (` 1 + τ ` )` 3 , -15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )` , 30` (` 1 + τ ` )` 2 `]`

For τ=1/2, [39, 13, 39, 27, 13, 36] . FixedPtCheck, [39, 13, 39, 27, 13, 36]

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, 3, 4, 6}},   net cycles: 1 .    order:   4

[y1, 0, y2, y3, 0, y4]  

See Matrices
 

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

[y2, y1, 0, 0, y3, y4]  

See Matrices
 

 » SYNC'D 37/512 , 0.07226562500

 
23 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -30` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [325, 637, 663, 51, 559, 204] . FixedPtCheck, [325, 637, 663, 51, 559, 204]

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 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: {{2, 3, 5}},   net cycles: 0 .    order:   3

[y3, y4, y5, 0, y1, y2]  

See Matrices
 

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

[y2, y1, 0, y3, y4, y5]  

See Matrices
 

 » SYNC'D 907/8192 , 0.1107177734

 
24 . Coloring, {3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 15` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )`` (` 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ ` )` , -30` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )``]`

For τ=1/2, [133, 357, 175, 75, 301, 100] . FixedPtCheck, [133, 357, 175, 75, 301, 100]

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 5 4 vs 4

See Matrix for A+τΔ

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

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

See Matrix
 

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

 

  p' = s 2 - s 4   p = s 2 - s 4

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

[y1, 0, y2, 0, y3, y4]  

See Matrices
 

 » SYNC'D 51/2048 , 0.02490234375

 
25 . Coloring, {3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 3 + τ 2 ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , -15` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 15` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` 2 , -15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 30` (` - 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-325, -637, -663, -153, -559, -204] . FixedPtCheck, [325, 637, 663, 153, 559, 204]

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 4 vs 4

See Matrix for A+τΔ

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

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

[y1, y2, y3, y5, y4, 0]  

See Matrices
 

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

[y1, y2, 0, 0, y3, y4]  

See Matrices
 

 » SYNC'D 397/1024 , 0.3876953125

 
26 . Coloring, {4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `15` (` 5 - τ + 3τ 2 + τ 3 ` )`` (` - 1 + τ ` )`` (` 3 + τ ` )` , 15` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )`` (` - 5 + τ 2 ` )` , 15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , -15` (` 1 + τ ` )` 3 ` (` 5 - 2τ + τ 2 ` )` , -15` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -30` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [-301, -133, -357, -459, -175, -612] . FixedPtCheck, [301, 133, 357, 459, 175, 612]

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 3 vs 3

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

[0, 0, y1, y2, y3, y4]  

See Matrices
 

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

[y2, y1, 0, 0, y3, 0]  

See Matrices
 

 » SYNC'D 23/64 , 0.3593750000

 
27 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ 2 ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` - 5 - τ - 3τ 2 + τ 3 ` )`` (` 3 + τ 2 ` )` , 15` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , 15` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )` 2 ` (` 1 + τ ` )` , 15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -30` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-663, -637, -741, -57, -325, -228] . FixedPtCheck, [663, 637, 741, 57, 325, 228]

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, 2, 3}},   net cycles: 0 .    order:   3

[y1, y2, y3, 0, 0, y4]  

See Matrices
 

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

[y2, y1, 0, y3, y4, y5]  

See Matrices
 

 » SYNC'D 109/1024 , 0.1064453125

 
28 . Coloring, {2, 3, 4, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 15` (` 1 + τ ` )`` (` 3 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 15` (` 1 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 30` (` 5 - τ + 3τ 2 + τ 3 ` )``]`

For τ=1/2, [343, 357, 301, 129, 175, 172] . FixedPtCheck, [343, 357, 301, 129, 175, 172]

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: {{4, 6}, {1, 2, 3}},   net cycles: 2 .    order:   6

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

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

[y2, 0, y1, 0, y4, y3]  

See Matrices
 

 » SYNC'D 315/4096 , 0.07690429688

 
29 . Coloring, {2, 3, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ 2 ` )`` (` 5 - 2τ + τ 2 ` )`` (` 1 + τ ` )` , 15` (` 3 + τ 2 ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 15` (` - 5 + τ 2 ` )`` (` 3 + τ 2 ` )`` (` 1 + τ ` )` , -15` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2 , 15` (` - 1 + τ ` )`` (` 3 + τ 2 ` )`` (` 5 + 2τ + τ 2 ` )` , -30` (` - 5 + τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-663, -637, -741, -171, -325, -228] . FixedPtCheck, [663, 637, 741, 171, 325, 228]

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, 2, 3}},   net cycles: 0 .    order:   3

[y1, y2, y3, y4, 0, 0]  

See Matrices
 

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

[y4, y3, 0, 0, y2, y1]  

See Matrices
 

 » SYNC'D 19/64 , 0.2968750000

 
30 . Coloring, {2, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )` , 15` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` , -15` (` - 1 + τ ` )`` (` 3 + τ ` )`` (` 1 + τ ` )` , 15` (` 1 + τ ` )` 3 , 15` (` - 1 + τ ` )` 2 ` (` 3 + τ ` )` , 30` (` 1 + τ ` )` 2 `]`

For τ=1/2, [21, 7, 21, 27, 7, 36] . FixedPtCheck, [21, 7, 21, 27, 7, 36]

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 3 vs 3

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, 0, y4]  

See Matrices
 

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

[y1, y2, 0, 0, y3, 0]  

See Matrices
 

 » SYNC'D 3/32 , 0.09375000000

 
31 . Coloring, {3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , -15` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 15` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 1 + τ ` )` 2 ` (` 5 - 2τ + τ 2 ` )` , 15` (` 3 + τ ` )`` (` 5 - τ + 3τ 2 + τ 3 ` )` , 30` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )``]`

For τ=1/2, [175, 343, 357, 153, 301, 204] . FixedPtCheck, [175, 343, 357, 153, 301, 204]

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: {{4, 6}, {2, 3, 5}},   net cycles: 2 .    order:   6

See Matrix
 

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

 

  p = - s - s 2 + s 4 + s 5

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

[y3, y4, 0, 0, y2, y1]  

See Matrices
 

 » SYNC'D 711/4096 , 0.1735839844

 
32 . Coloring, {2, 3, 4, 5, 6}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `-15` (` 3 + τ ` )`` (` 1 + τ ` )`` (` 5 - 2τ + τ 2 ` )` , 15` (` 3 + τ ` )`` (` - 5 - τ - 3τ 2 + τ 3 ` )` , 15` (` 3 + τ ` )`` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` , 15` (` - 5 + τ 2 ` )`` (` 1 + τ ` )` 2 , 15` (` 3 + τ ` )`` (` - 1 + τ ` )`` (` 5 + 2τ + τ 2 ` )` , 30` (` - 5 + τ 2 ` )`` (` 1 + τ ` )``]`

For τ=1/2, [-357, -343, -399, -171, -175, -228] . FixedPtCheck, [357, 343, 399, 171, 175, 228]

det(A + τ Δ) =   1` (` 1 + 3τ 2 ` )`` (` - 1 + τ ` )`` (` 1 + τ ` )` 2
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: {{4, 6}, {1, 2, 3}},   net cycles: 2 .    order:   6

See Matrix
 

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

 

  p = s + s 2 - s 4 - s 5

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

[y4, y3, 0, 0, y2, y1]  

See Matrices
 

 » SYNC'D 45/512 , 0.08789062500

 
SUMMARY
Graph Type
NOT CC
ν(A)
 0
ν(Δ)
 1
π
 [3, 3, 3, 1, 3, 2]
Dbly Stoch
false

 
SANDWICH
Total 0
No .ColoringRank

 
RT GROUPS
Total 1
No .ColoringRankSolv
1 {} 5 Not Solvable

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
31 0 31 , 30 24 , 26 1 32 32