Bishop Graph

Author

Eric W. Weisstein
December 7, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/BishopGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BishopGraph.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Graphs.m

GraphicsRow[Table[StyleGraphs@BishopGraph[n,n],{n,5}],Dividers->All,ImageSize->500]

GraphData "Grid" (degenerate)

GraphicsRow[Table[StyleGraphs@GraphData[{"Bishop",{n,n}},"Graph","Grid"][[-1]]],{n,5}],Dividers->All,ImageSize->500]

GraphData default

GraphicsRow[Table[StyleGraphs@GraphData[{"Bishop",{n,n}}],{n,5}],Dividers->All,ImageSize->500]

Special cases

With[{cl="Bishop"},TextGrid[Sort[{cl/.GraphData[#,"NotationRules"],GraphData[#,"#"]}&/@DeleteCases[GraphData[cl],{cl,_}]],Dividers->All]]

Embedding construction

(#->SetProperty[GraphData[{"Bishop",#}],VertexCoordinates->Automatic])&/@{{3,3},{3,4},{3,5},{4,4}}

With[{cl="Bishop"},SlideView[{Column[Union@{#,{cl,cl/.GraphData[#,"NotationRules"]}},Alignment->Center],GetGraphData[#,"File"],Grid[Transpose[{GraphData[#,"Graph","All"],GraphData[#,"EmbeddingClasses"]}],Dividers->All]}&/@Sort@GraphData@cl]
]

Embeddings

<<MathWorld`Graphs`

{"Bishop", {4, 9}}

GraphData[{"Bishop",{4,9}},"StandardName"]

Primary

GraphData[{"Bishop",{4,9}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Bishop",{4,9}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Bishop",{4,9}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Bishop",{4,9}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Bishop",{4,9}},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Bishop",{4,9}},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Bishop",{4,9}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Bishop",{4,9}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Bishop",{4,9}},"Planar"]

GraphData[{"Bishop",{4,9}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Bishop",{4,9}},{"RCN","CN"}]

GraphData[{"Bishop",{4,9}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Bishop",{4,9}}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Bishop",{4,9}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Bishop",{4,9}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Bishop",{4,9}}],"CrossingNumber"->cn,RandomSeed->i],{i,10}]//Union

GraphData[{"Bishop",{4,9}},"ConnectedComponents","Name"]

g0=GraphData[{"BlackBishop",{4,9}},"Graph","MinimalCrossing"][[1]]

g=MyGraphUnion[g0,g0]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Bishop",{4,9}},"Circulant"]

GraphData[{"Bishop",{4,9}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Bishop",{4,9}},"Halin"]

GraphData[{"Bishop",{4,9}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Bishop",{4,9}},"LCF"]

GraphData[{"Bishop",{4,9}},"HamiltonianCycleCount"]

GraphData[{"Bishop",{4,9}},"LCFSignature"]

GraphData[{"Bishop",{4,9}},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"Bishop",{4,9}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Bishop",{4,9}}],Debug->True]//Timing

GraphData[{"Bishop",{4,9}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Bishop",{4,9}},"VertexList"],GraphData[{"Bishop",{4,9}},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Bishop",{4,9}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Bishop",{4,9}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData[{"Bishop",{4,9}},"Matchstick"]

GraphData[{"Bishop",{4,9}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Bishop",{4,9}},"Edges"]]

3D All

GraphData[{"Bishop",{4,9}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Bishop",{4,9}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Bishop",{4,9}},"Polyhedral"]

GraphData[{"Bishop",{4,9}},"Graph",{"3D","Polyhedron"}]

{"Bishop", {4, 10}}

GraphData[{"Bishop",{4,10}},"StandardName"]

Primary

GraphData[{"Bishop",{4,10}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Bishop",{4,10}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Bishop",{4,10}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Bishop",{4,10}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Bishop",{4,10}},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Bishop",{4,10}},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Bishop",{4,10}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Bishop",{4,10}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Bishop",{4,10}},"Planar"]

GraphData[{"Bishop",{4,10}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Bishop",{4,10}},{"RCN","CN"}]

GraphData[{"Bishop",{4,10}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Bishop",{4,10}}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Bishop",{4,10}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Bishop",{4,10}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Bishop",{4,10}}],"CrossingNumber"->cn,RandomSeed->i],{i,10}]//Union

g0=GraphData[{"BlackBishop",{4,10}},"Graph","MinimalCrossing"][[1]]

g=MyGraphUnion[g0,g0]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Bishop",{4,10}},"Circulant"]

GraphData[{"Bishop",{4,10}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Bishop",{4,10}},"Halin"]

GraphData[{"Bishop",{4,10}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Bishop",{4,10}},"LCF"]

GraphData[{"Bishop",{4,10}},"HamiltonianCycleCount"]

GraphData[{"Bishop",{4,10}},"LCFSignature"]

GraphData[{"Bishop",{4,10}},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"Bishop",{4,10}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Bishop",{4,10}}],Debug->True]//Timing

GraphData[{"Bishop",{4,10}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Bishop",{4,10}},"VertexList"],GraphData[{"Bishop",{4,10}},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Bishop",{4,10}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Bishop",{4,10}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData[{"Bishop",{4,10}},"Matchstick"]

GraphData[{"Bishop",{4,10}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Bishop",{4,10}},"Edges"]]

3D All

GraphData[{"Bishop",{4,10}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Bishop",{4,10}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Bishop",{4,10}},"Polyhedral"]

GraphData[{"Bishop",{4,10}},"Graph",{"3D","Polyhedron"}]

{"Bishop", {5, 5}}

GraphData[{"Bishop",{5,5}},"StandardName"]

Primary

GraphData[{"Bishop",{5,5}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Bishop",{5,5}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Bishop",{5,5}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Bishop",{5,5}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Bishop",{5,5}},"Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Bishop",{5,5}},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"Bishop",{5,5}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Bishop",{5,5}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"Bishop",{5,5}},"Planar"]

GraphData[{"Bishop",{5,5}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Bishop",{5,5}},{"RCN","CN"}]

GraphData[{"Bishop",{5,5}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Bishop",{5,5}}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Bishop",{5,5}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Bishop",{5,5}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Bishop",{5,5}}],"CrossingNumber"->cn,RandomSeed->i],{i,10}]//Union

cc=GraphData[{"Bishop",{5,5}},"ConnectedComponents","Name"]

gs=GraphData[#,"Graph","MinimalCrossing"][[1]]&/@cc

g=MyGraphUnion@@gs

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Bishop",{5,5}},"Circulant"]

GraphData[{"Bishop",{5,5}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"Bishop",{5,5}},"Halin"]

GraphData[{"Bishop",{5,5}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Bishop",{5,5}},"LCF"]

GraphData[{"Bishop",{5,5}},"HamiltonianCycleCount"]

GraphData[{"Bishop",{5,5}},"LCFSignature"]

GraphData[{"Bishop",{5,5}},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"Bishop",{5,5}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Bishop",{5,5}}],Debug->True]//Timing

GraphData[{"Bishop",{5,5}},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Bishop",{5,5}},"VertexList"],GraphData[{"Bishop",{5,5}},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"Bishop",{5,5}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Bishop",{5,5}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData[{"Bishop",{5,5}},"Matchstick"]

GraphData[{"Bishop",{5,5}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Bishop",{5,5}},"Edges"]]

3D All

GraphData[{"Bishop",{5,5}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Bishop",{5,5}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Bishop",{5,5}},"Polyhedral"]

GraphData[{"Bishop",{5,5}},"Graph",{"3D","Polyhedron"}]

Properties m×n

Properties n×n

Acyclic

seq=Table[GraphData[{"Bishop",{n,n}},"Acyclic"],{n,10}]

AdjacencyMatrixCount

seq=Table[GraphData[{"Bishop",{n,n}},"AdjacencyMatrixCount"],{n,10}]

Table[(n^2)!/Piecewise[{{1,n==1}},(4^(-1+n)) (3+(((-1))^(1+n)))],{n,10}]

Arboricity

seq=Table[GraphData[{"Bishop",{n,n}},"Arboricity"],{n,2,10}]

ArcTransitive

seq=Table[GraphData[{"Bishop",{n,n}},"ArcTransitive"],{n,10}]

ArcTransitivity

seq=Table[GraphData[{"Bishop",{n,n}},"ArcTransitivity"],{n,10}]

AutomorphismCount

seq=Table[GraphData[{"Bishop",{n,n}},"AutomorphismCount"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

BalabanIndex

seq=Table[GraphData[{"Bishop",{n,n}},"BalabanIndex"],{n,10}]

Bandwidth

seq=Table[GraphData[{"Bishop",{n,n}},"Bandwidth"],{n,10}]

BridgeCount

seq=Table[GraphData[{"Bishop",{n,n}},"BridgeCount"],{n,10}]

BurningNumber

seq=Table[GraphData[{"Bishop",{n,n}},"BurningNumber"],{n,10}]

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"CharacteristicPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Chordal

seq=Table[GraphData[{"Bishop",{n,n}},"Chordal"],{n,10}]

ChordCount

seq=Table[GraphData[{"Bishop",{n,n}},"ChordCount"],{n,10}]

Chordless

seq=Table[GraphData[{"Bishop",{n,n}},"Chordless"],{n,10}]

ChordlessCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"ChordlessCycleCount"],{n,10}]

FindSequenceFunction[seq,n]

Chords

seq=Table[GraphData[{"Bishop",{n,n}},"Chords"],{n,10}]

ChromaticInvariant

seq=Table[GraphData[{"Bishop",{n,n}},"ChromaticInvariant"],{n,10}]

ChromaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"ChromaticNumber"],{n,10}]

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"ChromaticPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

FactorialPower[x,2]//FunctionExpand

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"ChromaticPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"Bishop",{n,n}},"Circumference"],{n,10}]

seq=Table[Circumference[BishopGraph[n,n]],{n,30}]

FindLinearRecurrence[Drop[seq,2]]

FindSequenceFunction[Drop[seq,2],n-2]

gf=(x^3)FindGeneratingFunction[Drop[seq,2],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["Web"])

CliqueCount

A183156

seq=Table[GraphData[{"Bishop",{n,n}},"CliqueCount"],{n,10}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

CliqueCoveringNumber

seq=Table[GraphData[{"Bishop",{n,n}},"CliqueCoveringNumber"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

CliqueNumber

seq=Table[GraphData[{"Bishop",{n,n}},"CliqueNumber"],{n,10}]

CliquePolynomial

A183157

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"CliquePolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

p=.

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"ComplementOddChordlessCycleCount"],{n,10}]

ConnectedComponentCount

seq=Table[GraphData[{"Bishop",{n,n}},"ConnectedComponentCount"],{n,10}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"Bishop",{n,n}},"ConnectedDominatingSetCount"],{n,10}]

ConnectedDominationNumber

seq=Table[GraphData[{"Bishop",{n,n}},"ConnectedDominationNumber"],{n,10}]

ConnectedDominationPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"ConnectedDominationPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

A291595

seq=Table[GraphData[{"Bishop",{n,n}},"ConnectedInducedSubgraphCount"],{n,10}]

seq={1,6,35,336,8095,389502,41541383,7705628640,3104759041723};

FindLinearRecurrence[seq]

FindLinearRecurrence[Rest@seq]

gf=x FindGeneratingFunction[seq,x]

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"CharacteristicPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,6}]

FullSimplify[poly2-poly]

CrossingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"CrossingNumber"],{n,10}]

CycleCount

A234636

seq=Table[GraphData[{"Bishop",{n,n}},"CycleCount"],{n,10}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

gf=(x^3)FindGeneratingFunction[seq,x]

CyclePolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"CyclePolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,{3,6}],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"Bishop",{n,n}},"Cyclic"],{n,10}]

CyclomaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"CyclomaticNumber"],{n,10}]

FindSequenceFunction[Drop[seq,2],n]//Factor

DetourIndex

seq=Table[GraphData[{"Bishop",{n,n}},"DetourIndex"],{n,10}]

DetourMatrix

MatrixForm/@Table[GraphData[{"Bishop",{n,n}},"DetourMatrix"],{n,10}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,10}]

DetourPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"DetourPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Diameter

seq=Table[GraphData[{"Bishop",{n,n}},"Diameter"],{n,10}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"Bishop",{n,n}},"DistanceMatrix"],{n,10}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,10}]

DistancePolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"DistancePolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

DistinguishingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"DistinguishingNumber"],{n,10}]

DomaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"DomaticNumber"],{n,10}]

DominatingSetCount

A295898

seq=Table[GraphData[{"Bishop",{n,n}},"DominatingSetCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq]

DominationNumber

seq=Table[GraphData[{"Bishop",{n,n}},"DominationNumber"],{n,10}]

DominationPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"DominationPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"EdgeChromaticNumber"],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

EdgeConnectivity

Table[GraphData[{"Bishop",{n,n}},"EdgeConnectivity"],{n,10}]

EdgeCount

seq=Table[GraphData[{"Bishop",{n,n}},"EdgeCount"],{n,10}]

FindSequenceFunction[seq,n]//Factor

EdgeCoverCount

seq=Table[GraphData[{"Bishop",{n,n}},"EdgeCoverCount"],{n,10}]

EdgeCoverNumber

seq=Table[GraphData[{"Bishop",{n,n}},"EdgeCoverNumber"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

x FindGeneratingFunction[seq,x]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"EdgeCoverPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"EulerianCycleCount"],{n,10}]

FaceCount

seq=Table[GraphData[{"Bishop",{n,n}},"FaceCount"],{n,10}]

FlowPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"FlowPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[Drop[DeleteMissing@poly,2]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

FractionalChromaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"FractionalChromaticNumber"],{n,10}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Bishop",{n,n}},"FractionalEdgeChromaticNumber"],{n,10}]

FindSequenceFunction[Drop[seq,1],n-1]//FullSimplify

Girth

seq=Table[GraphData[{"Bishop",{n,n}},"Girth"],{n,10}]

Gonality

seq=Table[GraphData[{"Bishop",{n,n}},"Gonality"],{n,10}]

HamiltonConnected

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonConnected"],{n,10}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonDecompositionCount"],{n,10}]

Hamiltonian

Table[GraphData[{"Bishop",{n,n}},"Hamiltonian"],{n,10}]

HamiltonianCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonianCycleCount"],{n,10}]

HamiltonianNumber

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonianNumber"],{n,10}]

HamiltonianPathCount

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonianPathCount"],{n,10}]

HamiltonianWalkCount

seq=Table[GraphData[{"Bishop",{n,n}},"HamiltonianWalkCount"],{n,10}]

HararyIndex

seq=Table[GraphData[{"Bishop",{n,n}},"HararyIndex"],{n,10}]

seq=Table[HararyIndex[BishopGraph[n,n]],{n,20}]

FindLinearRecurrence[Rest@seq]

FindSequenceFunction[seq,n]//FullSimplify[#,n∈Integers]&//Factor

Table[(1/48) (3(1-(-1)^n) +8 n-36 (n^2)+16  (n^3)+6 (n^4)),{n,10}]

HexagonCount

seq=Table[GraphData[{"Bishop",{n,n}},"HexagonCount"],{n,10}]

seq=Table[GraphCycleCount[BishopGraph[n,n],6],{n,30}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=(x^4) FindGeneratingFunction[Drop[seq,3],x]

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"IdiosyncraticPolynomial"][x,y],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"Bishop",{n,n}},"IndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

IndependencePolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"IndependencePolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"Bishop",{n,n}},"IndependenceRatio"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]

IndependentEdgeSetCount

A287248

seq=Table[GraphData[{"Bishop",{n,n}},"IndependentEdgeSetCount"],{n,10}]

black={2,12,130,9492,1166928,1431128744,2907639077764,76670800431934272,3341096345926174809912,2311650738313947870105792416,2645105778378736719464340469683304,56641723029988800376624313271476598959936};

white={2,7,130,4207,1166928,541240697,2907639077764,25886034363696809,3341096345926174809912,713738790204487208874991935,2645105778378736719464340469683304,16238808907214611705432043192158547965751};

seq=Prepend[black white,1]

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

IndependentVertexSetCount

A201862

seq=Table[GraphData[{"Bishop",{n,n}},"IndependentVertexSetCount"],{n,10}]

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,n-i}]

Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,n+i-k}]

knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!*Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,n-i}]*Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);

Table[1+Sum[knbishops[k,n],{k,2n-1}],{n,25}]

IntersectionNumber

seq=Table[GraphData[{"Bishop",{n,n}},"IntersectionNumber"],{n,10}]

IrredundantSetCount

A290714

Table[GraphData[{"Bishop",{n,n}},"IrredundantSetCount"],{n,10}]

KirchhoffIndex

seq=Table[GraphData[{"Bishop",{n,n}},"KirchhoffIndex"],{n,10}]

seq=Table[KirchhoffIndex[BishopGraph[n,n]],{n,20}]

KirchhoffSumIndex

seq=Table[GraphData[{"Bishop",{n,n}},"KirchhoffSumIndex"],{n,10}]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Bishop",{n,n}},"LaplacianMatrix"],{n,10}]

LaplacianPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"LaplacianPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"Bishop",{n,n}},"LeafCount"],{n,10}]

LongestCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"LongestCycleCount"],{n,10}]

LovaszNumber

seq=Table[GraphData[{"Bishop",{n,n}},"LovaszNumber"],{n,10}]

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"MatchingGeneratingPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

coef=Factor/@FindLinearRecurrence[Rest@DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"MatchingNumber"],{n,10}]

Table[Floor[n^2/2],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

MatchingPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"MatchingPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"Bishop",{n,n}},"MaximalCliqueCount"],{n,10}]

FindSequenceFunction[seq,n]

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"Bishop",{n,n}},"MaximalIndependentEdgeSetCount"],{n,10}]

MaximalIndependentVertexSetCount

A146304

seq=Table[GraphData[{"Bishop",{n,n}},"MaximalIndependentVertexSetCount"],{n,10}]

M[sig_List,n_,k_,d_,x_]:=M[sig,n,k,d,x]=If[n==0,Boole[k==0],If[k>0,k*x*M[sig,n-1,k-1,d,x],0]+If[k<n&&sig[[n]]>d,(sig[[n]]-d)*x*M[sig,n-1,k,d+1,x],0]+If[k+sig[[n]]-d<n,M[sig,n-1,k+sig[[n]]-d,sig[[n]],x],0]];
Q[sig_List,x_]:=M[sig,Length[sig],0,0,x];
Bishop[n_,white_]:=Table[n-i+If[white==1,1-Mod[i,2],Mod[i,2]],{i,1,n-If[white==1,Mod[n,2],1-Mod[n,2]]}]
a[n_]:=Q[Bishop[n,0],1]*Q[Bishop[n,1],1];

seq=Array[a,20]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

MaximumCliqueCount

seq=Table[GraphData[{"Bishop",{n,n}},"MaximumCliqueCount"],{n,10}]

FindSequenceFunction[seq,n]

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

MaximumIndependentEdgeSetCount

A286070

seq=Table[GraphData[{"Bishop",{n,n}},"MaximumIndependentEdgeSetCount"],{n,10}]

FindLinearRecurrence[Rest@DeleteMissing@seq]

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Bishop",{n,n}},"MaximumIndependentVertexSetCount"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]

MaximumLeafNumber

seq=Table[GraphData[{"Bishop",{n,n}},"MaximumLeafNumber"],{n,10}]

MaximumVertexDegree

A042948

seq=Table[GraphData[{"Bishop",{n,n}},"MaximumVertexDegree"],{n,10}]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[(1/2) ((4n-5)-(((-1))^n)),{n,20}]

Table[n-1+2Floor[(n-1)/2],{n,20}]

MeanDistance

seq=Table[GraphData[{"Bishop",{n,n}},"MeanDistance"],{n,10}]

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"Bishop",{n,n}},"MinimalConnectedDominatingSetCount"],{n,10}]

MinimalDominatingSetCount

A295899

seq=Table[GraphData[{"Bishop",{n,n}},"MinimalDominatingSetCount"],{n,10}]

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumCliqueCoveringCount"],{n,10}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumConnectedDominatingSetCount"],{n,10}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumDistinguishingLabelingCount"],{n,10}]

MinimumDominatingSetCount

A182333

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumDominatingSetCount"],{n,10}]

seq=Table[If[n==1,1,((2*Floor[n/4])!)^2/128*(n^5+3*n^4+n^3+35*n^2+38*n+2-(n^5-n^4-7*n^3-n^2-10*n-30)*(-1)^n-4*(n^3+2*n^2+n-4)*n*Cos[Pi*n/2]-2*(n^5+n^4-11*n^3-7*n^2-2*n+2)*Sin[Pi*n/2])],{n,1,25}]

Head/@Table[FindLinearRecurrence[Drop[seq,n]],{n,5}]

FindGeneratingFunction[seq,x]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumEdgeCoverCount"],{n,10}]

MinimumVertexDegree

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumVertexDegree"],{n,10}]

MolecularTopologicalIndex

seq=Table[GraphData[{"Bishop",{n,n}},"MolecularTopologicalIndex"],{n,10}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]//Factor

Nonhamiltonian

Table[GraphData[{"Bishop",{n,n}},"Nonhamiltonian"],{n,10}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"Bishop",{n,n}},"NormalizedLaplacianMatrix"],{n,10}]

OddChordlessCycleCount

seq=Table[GraphData[{"Bishop",{n,n}},"OddChordlessCycleCount"],{n,10}]

PathCount

A286124

seq=Table[GraphData[{"Bishop",{n,n}},"PathCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]//Factor

gf=(x^3) FindGeneratingFunction[DeleteMissing@seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>2]

PathPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"PathPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

PentagonCount

seq=Table[GraphData[{"Bishop",{n,n}},"PentagonCount"],{n,10}]

seq=Table[GraphCycleCount[BishopGraph[n,n],5],{n,25}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

Table[(1/120) (-15+15 (((-1))^n)-152 n+492 (n^2)-500 (n^3)+250 (n^4)-68 (n^5)+8 (n^6)),{n,10}]

gf=x FindGeneratingFunction[seq,x]

Perfect

Table[GraphData[{"Bishop",{n,n}},"Perfect"],{n,10}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"ProjectivePlaneCrossingNumber"],{n,10}]

Radius

seq=Table[GraphData[{"Bishop",{n,n}},"Radius"],{n,10}]

RankPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"RankPolynomial"][x,y],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"RectilinearCrossingNumber"],{n,10}]

ReliabilityPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"ReliabilityPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"SigmaPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

SpanningTreeCount

seq=Table[GraphData[{"Bishop",{n,n}},"SpanningTreeCount"],{n,10}]

SquareCount

seq=Table[GraphData[{"Bishop",{n,n}},"SquareCount"],{n,10}]

FindSequenceFunction[seq,n]

StabilityIndex

seq=Table[GraphData[{"Bishop",{n,n}},"StabilityIndex"],{n,10}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"Bishop",{n,n}},"TopologicalIndex"],{n,10}]

ToroidalCrossingNumber

seq=Table[GraphData[{"Bishop",{n,n}},"ToroidalCrossingNumber"],{n,10}]

Traceable

Table[GraphData[{"Bishop",{n,n}},"Traceable"],{n,10}]

TriangleCount

seq=Table[GraphData[{"Bishop",{n,n}},"TriangleCount"],{n,10}]

FindSequenceFunction[seq,n]//Factor

TuttePolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"TuttePolynomial"][x,y],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"Bishop",{n,n}},"Untraceable"],{n,10}]

VertexConnectivity

Table[GraphData[{"Bishop",{n,n}},"VertexConnectivity"],{n,10}]

VertexCount

seq=Table[GraphData[{"Bishop",{n,n}},"VertexCount"],{n,10}]

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

seq=Table[GraphData[{"Bishop",{n,n}},"VertexCoverNumber"],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

Factor/@FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

VertexCoverPolynomial

(poly=Factor[Table[GraphData[{"Bishop",{n,n}},"VertexCoverPolynomial"],{n,k0=1,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>0]

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"Bishop",{n,n}},"VertexDegrees"],{n,10}]

VertexTransitive

Table[GraphData[{"Bishop",{n,n}},"VertexTransitive"],{n,10}]

WienerIndex

seq=Table[GraphData[{"Bishop",{n,n}},"WienerIndex"],{n,10}]

FindSequenceFunction[seq,n]

WienerSumIndex

seq=Table[GraphData[{"Bishop",{n,n}},"WienerSumIndex"],{n,10}]

FindSequenceFunction[seq,n]

GraphData

GraphData["Bishop"]

GraphDataClassString

GraphDataClassString["Bishop"]

GraphDataString

file=OpenAppend["~/bishop.m"]

Do[WriteString[file,GraphDataString[BishopGraph[n,n],
"StandardName"->{"Bishop",{n,n}},
"Name"->"("<>ToString[n]<>","<>ToString[n]<>")\[Hyphen]bishop graph",
"Information"->"BishopGraph",
"Classes"->{"Bishop"},
"NotationRules"->{"Bishop"->{n,n}},
"Skip"->{"ABCEnergy","ABCSpectralRadius","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","BetweennessCentralities","BurningNumber","CharacteristicPolynomial","Circumference","CliqueCoveringNumber","ConnectedDominatingSets","ConstructionNumber","ConnectedInducedSubgraphPolynomial","ConnectedIndicedSubgraphs","Curvatures","CyclePolynomial","CyclicEdgeConnectivity","DetourMatrix","DominatingSets","EigenvectorCentralities","Embeddings3D","Energy","Genus","GracefulLabeling","HadwigerNumber","HamiltonianPaths","HamiltonianWalk","HamiltonianWalks","HITSCentralities","IdiosyncraticPolynomial","IndependencePolynomial","IrredundacePolynomial","IrredundantSets","KatzCentralities","LinklesslyEmbeddable","LowerMatchingNumber","MatchingGeneratingPolynomial","MatchingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSet","MaximalIndependentVertexSets","MaximumIndependentEdgeSetCount","MaximumIndependentEdgeSets","MaximumIndependentVertexSet","MaximumIndependentVertexSetCount","MaximumIndependentVertexSets","MaximumIrredundantSets","MetricDimension","Meyniel","MinimumCoveringsByMaximalCliques","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","MinimumEdgeCuts","MinimumPathCovering","MinimumVertexCover","MinimumTotalDominatingSet","MinimumWeightFractionalEdgeColoring","MinimumWeightFractionalColoring","Minors","Outerplanar","PageRankCentralities","PathCoveringNumber","PathPolynomial","Paths","ProjectivePlanar","QuadraticEmbeddingConstant","RandicEnergy","RandicSpectralRadius","RectilinearCrossingCount","Skewness","SomborEnergy","SomborSpectralRadius","SpanningTreeCount","TotalDominationPolynomial","TuttePolynomial"},
Debug->True,
TimeConstraint->10
]],{n,{13}}]

Close[file]