New Graph

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

 

 

π = [1, 1, 1, 1, 1]

POSSIBLE RANKS

1 x 5

BASE DETERMINANT 725/4096, .1770019531

NullSpace of Δ

{1, 2, 3, 4, 5}

Nullspace of A

` det(A) = ` 1/16

 

 
1 . Coloring, {}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [225, 195, 67, 73, 105] . FixedPtCheck, [225, 195, 67, 73, 105]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 2}},   net cycles: 0 .    order:   2

[y3, y2, 0, 0, y1]  

See Matrices
 

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

[0, y3, y1, y2, 0]  

See Matrices
 

 » SYNC'D 3/16 , 0.1875000000

 
2 . Coloring, {2}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-15, -15, -13, -7, -15] . FixedPtCheck, [15, 15, 13, 7, 15]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y4, y2, y3, 0, y1]  

See Matrices
 

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

[y1, y2, y3, y4, 0]  

See Matrices
 

 » SYNC'D 9/64 , 0.1406250000

 
3 . Coloring, {3}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-195, -169, -67, -99, -91] . FixedPtCheck, [195, 169, 67, 99, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y1, y2, 0, y4, y3]  

See Matrices
 

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

[0, y2, y1, y3, y4]  

See Matrices
 

 » SYNC'D 31/256 , 0.1210937500

 
4 . Coloring, {4}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [195, 169, 97, 73, 91] . FixedPtCheck, [195, 169, 97, 73, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y4, y3, y2, 0, y1]  

See Matrices
 

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

[0, y1, y2, y3, y4]  

See Matrices
 

 » SYNC'D 11/64 , 0.1718750000

 
5 . Coloring, {5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [195, 225, 73, 67, 105] . FixedPtCheck, [195, 225, 73, 67, 105]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{1, 2}},   net cycles: 0 .    order:   2

[y3, y2, 0, 0, y1]  

See Matrices
 

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

[y1, 0, y2, y3, 0]  

See Matrices
 

 » SYNC'D 3/16 , 0.1875000000

 
6 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

`

Ω for A+τΔ : 
       `[ `1 , 1 , 1 , 1 , 1`]`

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

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

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

[1, 1, 1, 1, 1]

+              \ ;   -              \ ;   Δ

See Matrices

 
[0, 0, 0, 0, 0]
  p' = s   p' = s 3   p' = s 2   p = s

        S+              \ ;   S-              \ ;   NM
See Matrices

CmmCk true, true, true


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

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

See Matrix
 

[y1, y1, y1, y1, y1]

 

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

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

See Matrix
 

[y1, y1, y1, y1, y1]

 

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


  « NOT SYNC'D »

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

Range of {ΩΔi}: [0, 0, 0, 0, 0]

 
rank of M is 5 , rank of N is 5

M              N

$ [ [0, 1, 1, 1, 1] , [1, 0, 1, 1, 1] , [1, 1, 0, 1, 1] , [1, 1, 1, 0, 1] , [1, 1, 1, 1, 0] ] $     $ [ [0, 1, 1, 1, 1] , [1, 0, 1, 1, 1] , [1, 1, 0, 1, 1] , [1, 1, 1, 0, 1] , [1, 1, 1, 1, 0] ] $

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

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

τ= 5 , r'= 4/5

Ranges

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

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

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

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

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

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

` ` See level-5 partition graph.

`

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

 

 
7 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [13, 13, 15, 7, 13] . FixedPtCheck, [13, 13, 15, 7, 13]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y2, y1, y3, 0, y4]  

See Matrices
 

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

[y4, y3, 0, y1, y2]  

See Matrices
 

 » SYNC'D 27/128 , 0.2109375000

 
8 . Coloring, {2, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [105, 225, 187, 73, 195] . FixedPtCheck, [105, 225, 187, 73, 195]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[0, y1, y2, 0, y3]  

See Matrices
 

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

[y1, 0, y2, y3, 0]  

See Matrices
 

 » SYNC'D 11/32 , 0.3437500000

 
9 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-105, -91, -97, -99, -49] . FixedPtCheck, [105, 91, 97, 99, 49]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

See Matrix
 

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

 

  p = - s + s 3   p' = - s + s 3

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

[0, y1, y2, y3, y4]  

See Matrices
 

 » SYNC'D 9/256 , 0.03515625000

 
10 . Coloring, {3, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [169, 195, 73, 97, 91] . FixedPtCheck, [169, 195, 73, 97, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y2, y4, 0, y1, y3]  

See Matrices
 

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

[y1, 0, y2, y3, y4]  

See Matrices
 

 » SYNC'D 11/64 , 0.1718750000

 
11 . Coloring, {4, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-169, -195, -99, -67, -91] . FixedPtCheck, [169, 195, 99, 67, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y2, y1, y3, 0, y4]  

See Matrices
 

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

[y2, 0, y1, y4, y3]  

See Matrices
 

 » SYNC'D 31/256 , 0.1210937500

 
12 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-7, -7, -15, -13, -7] . FixedPtCheck, [7, 7, 15, 13, 7]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[y1, y2, y3, y4, 0]  

See Matrices
 

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

[y1, y2, 0, y3, y4]  

See Matrices
 

 » SYNC'D 9/64 , 0.1406250000

 
13 . Coloring, {2, 3, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-91, -195, -187, -163, -169] . FixedPtCheck, [91, 195, 187, 163, 169]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[0, y4, y1, y2, y3]  

See Matrices
 

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

[y4, 0, y3, y2, y1]  

See Matrices
 

 » SYNC'D 25/128 , 0.1953125000

 
14 . Coloring, {2, 4, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [91, 195, 201, 73, 169] . FixedPtCheck, [91, 195, 201, 73, 169]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

[0, y3, y2, 0, y1]  

See Matrices
 

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

[y1, 0, 0, y3, y2]  

See Matrices
 

 » SYNC'D 13/64 , 0.2031250000

 
15 . Coloring, {3, 4, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [-91, -105, -99, -97, -49] . FixedPtCheck, [91, 105, 99, 97, 49]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

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

See Matrix
 

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

 

  p' = s - s 3   p = s - s 3

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

[y1, 0, y3, y4, y2]  

See Matrices
 

 » SYNC'D 9/256 , 0.03515625000

 
16 . Coloring, {2, 3, 4, 5}

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

` See graph

` ` See pair graph

`

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

For τ=1/2, [49, 105, 201, 163, 91] . FixedPtCheck, [49, 105, 201, 163, 91]

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

See Matrix for A+τΔ

  Check x AllOnes:   [1, 1, 1, 1, 1]

Omega Rank for R :  cycles: {{3, 4}},   net cycles: 0 .    order:   2

[0, y2, y3, y1, 0]  

See Matrices
 

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

[y3, 0, 0, y1, y2]  

See Matrices
 

 » SYNC'D 11/32 , 0.3437500000

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

 
SANDWICH
Total 0
No .ColoringRank

 
RT GROUPS
Total 1
No .ColoringRankSolv
1 {2, 3} 5 ["group", Not Solvable]

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
15 0 15 , 15 13 , 15 1 16 16