New Graph

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

π = [3, 5, 6, 7]

POSSIBLE RANKS

1 x 21
3 x 7

BASE DETERMINANT 17/64, .2656250000

NullSpace of Δ

{1, 2, 3, 4}

Nullspace of A

` det(A) = ` -1/8

1 . Coloring, {}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-21` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 21` (` 1 + τ ` )`` (` - 5 + τ ² ` )` , -42` (` 3 + τ ² ` )` , 21` (` 7 + 4τ + τ ² ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-39, -57, -52, -37] . FixedPtCheck, [39, 57, 52, 37]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	3 vs 3	2 vs 2

See Matrix for A+τΔ

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

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

[y₃, y₁, y₂, 0]

See Matrices

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

[0, 0, y₁, y₂]

See Matrices

» SYNC'D 7/16 , 0.4375000000

2 . Coloring, {2}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` 1 + τ ` )`` (` 3 + τ ` )`` (` - 1 + τ ` )` , 21` (` 1 + τ ` )`` (` - 5 + τ ² ` )` , 42` (` 3 + τ ` )`` (` - 1 + τ ` )` , -21` (` 7 - τ + τ ² + τ ³ ` )``]`

For τ=1/2, [-21, -57, -28, -55] . FixedPtCheck, [21, 57, 28, 55]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	3 vs 4	4 vs 4	3 vs 3	2 vs 2

See Matrix for A+τΔ

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

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

[y₁, y₃, 0, y₂]

See Matrices

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

[0, 0, y₁, y₂]

See Matrices

» SYNC'D 1/8 , 0.1250000000

3 . Coloring, {3}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `-21` (` 3 + τ ² ` )`` (` - 1 + τ ` )` , 21` (` 1 + τ ` )`` (` 5 - 2τ + τ ² ` )` , 42` (` 3 + τ ² ` )` , 21` (` 7 - τ + τ ² + τ ³ ` )``]`

For τ=1/2, [13, 51, 52, 55] . FixedPtCheck, [13, 51, 52, 55]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	3 vs 3	3 vs 3

See Matrix for A+τΔ

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

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

[0, y₃, y₁, y₂]

See Matrices

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

[y₁, 0, y₂, y₃]

See Matrices

» SYNC'D 7/32 , 0.2187500000

4 . Coloring, {4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` 1 + τ ` )` ² ` (` - 3 + τ ` )` , -21` (` 5 - τ + 3τ ² + τ ³ ` )` , 42` (` 1 + τ ` )`` (` - 3 + τ ` )` , 21` (` 7 + 4τ + τ ² ` )`` (` - 1 + τ ` )``]`

For τ=1/2, [-45, -43, -60, -37] . FixedPtCheck, [45, 43, 60, 37]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	3 vs 3	2 vs 2

See Matrix for A+τΔ

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

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

[y₁, y₂, y₃, 0]

See Matrices

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

[0, y₂, 0, y₁]

See Matrices

» SYNC'D 7/16 , 0.4375000000

5 . Coloring, {2, 3}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` 3 + τ ` )`` (` - 1 + τ ` )` ² , 21` (` 1 + τ ` )`` (` 5 - 2τ + τ ² ` )` , -42` (` 3 + τ ` )`` (` - 1 + τ ` )` , -21` (` - 7 - τ - τ ² + τ ³ ` )``]`

For τ=1/2, [7, 51, 28, 61] . FixedPtCheck, [7, 51, 28, 61]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	2 vs 2	3 vs 3

See Matrix for A+τΔ

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

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

[0, y₁, 0, y₂]

See Matrices

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

[y₃, 0, y₂, y₁]

See Matrices

» SYNC'D 7/16 , 0.4375000000

6 . Coloring, {2, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` 1 + τ ` )`` (` 3 + τ ² ` )` , 21` (` 5 - τ + 3τ ² + τ ³ ` )` , 42` (` 3 + τ ² ` )` , 21` (` 7 - τ + τ ² + τ ³ ` )``]`

For τ=1/2, [39, 43, 52, 55] . FixedPtCheck, [39, 43, 52, 55]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	3 vs 4	4 vs 4	4 vs 4	3 vs 3

See Matrix for A+τΔ

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

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

[y₁, y₂, y₄, y₃]

See Matrices

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

[0, y₁, y₂, y₃]

See Matrices

» SYNC'D 5/32 , 0.1562500000

7 . Coloring, {3, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` 1 + τ ` )`` (` - 1 + τ ` )`` (` - 3 + τ ` )` , -21` (` - 1 + τ ` )`` (` 5 + 2τ + τ ² ` )` , -42` (` 1 + τ ` )`` (` - 3 + τ ` )` , 21` (` 7 - τ + τ ² + τ ³ ` )``]`

For τ=1/2, [3, 5, 12, 11] . FixedPtCheck, [3, 5, 12, 11]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	3 vs 3	3 vs 3

See Matrix for A+τΔ

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

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

[0, y₃, y₁, y₂]

See Matrices

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

[y₁, y₂, 0, y₃]

See Matrices

» SYNC'D 1/8 , 0.1250000000

8 . Coloring, {2, 3, 4}

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

` See graph

` ` See pair graph

Ω for A+τΔ :
`[ `21` (` - 1 + τ ` )`` (` 3 + τ ² ` )` , 21` (` - 1 + τ ` )`` (` 5 + 2τ + τ ² ` )` , -42` (` 3 + τ ² ` )` , 21` (` - 7 - τ - τ ² + τ ³ ` )``]`

For τ=1/2, [-13, -25, -52, -61] . FixedPtCheck, [13, 25, 52, 61]

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

Δ-Rank	A+(1/2)Δ	A-(1/2)Δ	R	B
3 vs 3	4 vs 4	4 vs 4	3 vs 3	4 vs 4

See Matrix for A+τΔ

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

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

[0, y₁, y₃, y₂]

See Matrices

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

[y₄, y₃, y₂, y₁]

See Matrices

» SYNC'D 1/16 , 0.06250000000

SUMMARY

Graph Type
NOT CC

ν(A)
0

ν(Δ)
1

π
[3, 5, 6, 7]

Dbly Stoch
false

SUMMARY
Graph Type		NOT CC
ν(A)		0
ν(Δ)		1
π		[3, 5, 6, 7]
Dbly Stoch		false

SANDWICH
Total 0

No . Coloring Rank

SANDWICH		Total 0
No .	Coloring	Rank

RT GROUPS
Total 0

No . Coloring Rank Solv

RT GROUPS		Total 0
No .	Coloring	Rank	Solv

Δ-RANK'D SC'D !RK'D τ-RANK'D R/B RANK'D NOT SYNC'D Total Runs 2^n-1
8 0 6 , 8 8 , 8 0 8 8