White Bishop Graph

Author

Eric W. Weisstein
September 2, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Graphs.m

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

GraphData "Grid" (degenerate)

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

GraphData default

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

Curved edges

Table[GraphData[{"WhiteBishop",{n,n}},"Graph","All"][[-1]]/.g_Graph:>AdjacencyGraph[AdjacencyMatrix[g],EdgeShapeFunction->"Line",DirectedEdges->True,VertexCoordinates->GraphEmbedding[g]],{n,2,6}]

White vs Black

{BlackBishopGraph[3,5],WhiteBishopGraph[3,5]}

{BlackBishopGraph[3,4],WhiteBishopGraph[3,4]}

{BlackBishopGraph[4,6],WhiteBishopGraph[4,6]}

Special cases

Note: 1×1 white bishop graph is the null graph (1×1 chessboard is black)

Flatten[Table[{m,n}->RecognizeGraph@WhiteBishopGraph[m,n],{n,10},{m,n}]]

Grid[DeleteCases[Cases[Flatten[Table[{{m,n},WhiteBishopGraph[m,n]},{n,7},{m,n}],1]/.g_Graph:>(ToEntity[g]/._ToEntity:>{}),{i_,e_Entity}:>{i,CanonicalName[e]}],{_,{"BlackBishop"|"WhiteBishop",_}}],Dividers->All,Alignment->Left]

Embeddings

<<MathWorld`Graphs`

{"WhiteBishop", {3, 5}}

Names

GraphData[{"WhiteBishop",{3,5}},"StandardName"]

GraphData[{"WhiteBishop",{3,5}},"StandardNames"]

Primary

GraphData[{"WhiteBishop",{3,5}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"WhiteBishop",{3,5}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"WhiteBishop",{3,5}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"WhiteBishop",{3,5}}]]

Default embedding

GraphPlot[Graph[GraphData[{"WhiteBishop",{3,5}},"VertexList"],GraphData[{"WhiteBishop",{3,5}},"Edges"]]]

s=Sqrt[3]/2;
v={{0,0},{1,0},{1,1},{0,1},{1/2,1/2},{-s,1/2},{1+s,1/2}};
g=CenterGraph@Graph[Range[Length[v]],UndirectedEdge@@@{e[6,1,2,7,3,4,6],e[1,5,3],e[2,5,4]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

GraphPlot[Graph[GraphData[{"WhiteBishop",{3,5}},"VertexList"],GraphData[{"WhiteBishop",{3,5}},"Edges"]]]

v={{0,0},{1,0},{1,1/2},{1,1},{0,1},{0,1/2},{1/2,1/2}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,1,7,4],e[2,7,5]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

gs=UnlabelGraph/@{,}

Degenerate

GraphData[{"WhiteBishop",{3,5}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"WhiteBishop",{3,5}},"Graph","Degenerate"]

Configuration

GraphData[{"WhiteBishop",{3,5}},"Configuration"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"WhiteBishop",{3,5}},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"WhiteBishop",{3,5}},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"WhiteBishop",{3,5}},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"WhiteBishop",{3,5}},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"WhiteBishop",{3,5}},"Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingQ[#],GraphSymmetryGroup[#]}&/@GraphData[{"WhiteBishop",{3,5}},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

GraphData[{"WhiteBishop",{3,5}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"WhiteBishop",{3,5}},"Hamiltonian"]

GraphData[{"WhiteBishop",{3,5}},"HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData[{"WhiteBishop",{3,5}}],All][[All,All,1]]]

circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"WhiteBishop",{3,5}}],#]&/@hc

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

RotateGraph[,Pi]

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData[{"WhiteBishop",{3,5}},"Planar"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Planar"]//StyleGraphs

Planar construction

PlanarGraphEmbedding[GraphData[{"WhiteBishop",{3,5}}],Method->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData[{"WhiteBishop",{3,5}}]],10]

PlanarGraphEmbeddings

(plemb=PlanarGraphEmbeddings[GraphData[{"WhiteBishop",{3,5}}]])//Timing

Length[plemb]

MinimalCrossing

GraphData[{"WhiteBishop",{3,5}},{"RCN","CN"}]

GraphData[{"WhiteBishop",{3,5}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->86400])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,5}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

gs=IGLayoutTutte[g,"OuterFace"->#]&/@faces

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,5}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Select[gs,Max[OuterFace[#]]<=GraphData[{"WhiteBishop",{3,5}},"V"]&]

(gs=Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,5}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"WhiteBishop",{3,5}},"V"]&]])//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,5}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,5}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,5}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

RectilinearLocalCrossingCount[g]

MinimalLocalCrossing

GraphData[{"WhiteBishop",{3,5}},"LocalCrossingNumber"]

GraphData[{"WhiteBishop",{3,5}},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,5}}],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,5}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Select[Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,5}}],"CrossingNumber"->(cn=XXX),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,5}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Circulant

GraphData[{"WhiteBishop",{3,5}},"Circulant"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"WhiteBishop",{3,5}},"ComplementGraph","Name"]

GraphData[compl,"Graph","All"]

gs=GraphComplement/@%

Select[%,!DegenerateGraphEmbeddingQ[#]&]

GraphPlot@

v=Join[{{0,Sqrt[3]}},Subdivide[{-1,0},{-1/4,3/4},2],Subdivide[{1,0},{1/4,3/4},2]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[7,1,2,7,6,5,1,4,3,2],{4,5}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

RotateGraph[,{3,6}]

v=CirclePoints[7];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[7,2,4,6,1,4,7,3,6],e[1,5,2]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

gs=UnlabelGraph/@{,}

Line graph

GraphData[{"WhiteBishop",{3,5}},"Line"]

LineGraphQ[GraphData[{"WhiteBishop",{3,5}}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"WhiteBishop",{3,5}}&]

RootGraph[GraphData[{"WhiteBishop",{3,5}}]]

RecognizeGraph[%]

GraphData[roots[[1]],"Graph","All"]

gs=LineGraph/@%

Halin

GraphData[{"WhiteBishop",{3,5}},"Halin"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"WhiteBishop",{3,5}},"Rigid"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"WhiteBishop",{3,5}},"Flexible"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"WhiteBishop",{3,5}},"LCF"]

GraphData[{"WhiteBishop",{3,5}},"HamiltonianCycleCount"]

GraphData[{"WhiteBishop",{3,5}},"LCFSignature"]

GraphData[{"WhiteBishop",{3,5}},"LCFNotations"]

GraphData[{"WhiteBishop",{3,5}},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

UnitDistance

GraphData[{"WhiteBishop",{3,5}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"WhiteBishop",{3,5}}],Debug->True]//Timing

GraphData[{"WhiteBishop",{3,5}},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"WhiteBishop",{3,5}},"VertexList"],GraphData[{"WhiteBishop",{3,5}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"WhiteBishop",{3,5}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,5}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,5}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"WhiteBishop",{3,5}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,5}}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

gs={RotateGraph[,{9,5}],RotateGraph[,{8,7}],,,RotateGraph[,{8,2}]}

Matchstick

GraphData[{"WhiteBishop",{3,5}},"Matchstick"]

GraphData[{"WhiteBishop",{3,5}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"WhiteBishop",{3,5}},"Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"WhiteBishop",{3,5}},"Polyhedral"]

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

{"WhiteBishop", {3, 7}}

Names

GraphData[{"WhiteBishop",{3,7}},"StandardName"]

GraphData[{"WhiteBishop",{3,7}},"StandardNames"]

Primary

GraphData[{"WhiteBishop",{3,7}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"WhiteBishop",{3,7}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"WhiteBishop",{3,7}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"WhiteBishop",{3,7}}]]

Default embedding

GraphPlot[Graph[GraphData[{"WhiteBishop",{3,7}},"VertexList"],GraphData[{"WhiteBishop",{3,7}},"Edges"]],GraphLayout->"HighDimensionalEmbedding"]

s=Sqrt[3]/2;
v={{-1/2-s,1/2},{-1/2,0},{0,0},{1/2,0},{1/2+s,1/2},{-1/2,1/2},{1/2,1/2},{-1/2,1},{0,1},{1/2,1}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,10,9,8,1],e[2,6,8],e[4,7,10],e[3,7,9,6,3]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

GraphPlot[Graph[GraphData[{"WhiteBishop",{3,7}},"VertexList"],GraphData[{"WhiteBishop",{3,7}},"Edges"]],GraphLayout->"HighDimensionalEmbedding"]

v={{0,0},{1/2,0},{1,0},{1,1/2},{1,1},{1/2,1},{0,1},{0,1/2},{1/3,1/2},{2/3,1/2}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,8,1,9,7],e[3,10,5],e[2,10,6,9,2]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

gs=UnlabelGraph/@{,}

GraphPlot@RotateGraph[Graph[Graph[GraphData[{"WhiteBishop",{3,7}},"VertexList"],GraphData[{"WhiteBishop",{3,7}},"Edges"]],GraphLayout->"TutteEmbedding"],{9,5}]

RegionIntersection[Line[v[[{2,5}]]],Line[v[[{1,4}]]]]//FullSimplify

v=Join[CirclePoints[{1,Pi/2},8],{(1-(Sqrt[2]))#,0}&/@{-1,1}];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[6,9,{1,8},1,10,4],e[2,10,5,9,8]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

Degenerate

GraphData[{"WhiteBishop",{3,7}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"WhiteBishop",{3,7}},"Graph","Degenerate"]

Configuration

GraphData[{"WhiteBishop",{3,7}},"Configuration"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"WhiteBishop",{3,7}},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"WhiteBishop",{3,7}},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"WhiteBishop",{3,7}},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"WhiteBishop",{3,7}},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"WhiteBishop",{3,7}},"Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingQ[#],GraphSymmetryGroup[#]}&/@GraphData[{"WhiteBishop",{3,7}},"Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

GraphData[{"WhiteBishop",{3,7}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"WhiteBishop",{3,7}},"Hamiltonian"]

GraphData[{"WhiteBishop",{3,7}},"HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData[{"WhiteBishop",{3,7}}],All][[All,All,1]]]

circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"WhiteBishop",{3,7}}],#]&/@hc

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData[{"WhiteBishop",{3,7}},"Planar"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Planar"]//StyleGraphs

Planar construction

PlanarGraphEmbedding[GraphData[{"WhiteBishop",{3,7}}],Method->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData[{"WhiteBishop",{3,7}}]],10]

PlanarGraphEmbeddings

(plemb=PlanarGraphEmbeddings[GraphData[{"WhiteBishop",{3,7}}]])//Timing

Length[plemb]

MinimalCrossing

GraphData[{"WhiteBishop",{3,7}},{"RCN","CN"}]

GraphData[{"WhiteBishop",{3,7}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->86400])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,7}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

gs=IGLayoutTutte[g,"OuterFace"->#]&/@faces

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,7}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Select[gs,Max[OuterFace[#]]<=GraphData[{"WhiteBishop",{3,7}},"V"]&]

(gs=Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,7}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"WhiteBishop",{3,7}},"V"]&]])//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,7}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,7}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,7}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

RectilinearLocalCrossingCount[g]

MinimalLocalCrossing

GraphData[{"WhiteBishop",{3,7}},"LocalCrossingNumber"]

GraphData[{"WhiteBishop",{3,7}},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,7}}],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,7}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Select[Table[MinimalCrossingGraph[GraphData[{"WhiteBishop",{3,7}}],"CrossingNumber"->(cn=XXX),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData[{"WhiteBishop",{3,7}},"V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Circulant

GraphData[{"WhiteBishop",{3,7}},"Circulant"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"WhiteBishop",{3,7}},"ComplementGraph","Name"]

Line graph

GraphData[{"WhiteBishop",{3,7}},"Line"]

LineGraphQ[GraphData[{"WhiteBishop",{3,7}}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"WhiteBishop",{3,7}}&]

RootGraph[GraphData[{"WhiteBishop",{3,7}}]]

RecognizeGraph[%]

GraphData[roots[[1]],"Graph","All"]

gs=LineGraph/@%

g=CenterGraph@RotateGraph[,Pi/2]

Halin

GraphData[{"WhiteBishop",{3,7}},"Halin"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"WhiteBishop",{3,7}},"Rigid"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"WhiteBishop",{3,7}},"Flexible"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"WhiteBishop",{3,7}},"LCF"]

GraphData[{"WhiteBishop",{3,7}},"HamiltonianCycleCount"]

GraphData[{"WhiteBishop",{3,7}},"LCFSignature"]

GraphData[{"WhiteBishop",{3,7}},"LCFNotations"]

GraphData[{"WhiteBishop",{3,7}},"Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

UnitDistance

GraphData[{"WhiteBishop",{3,7}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"WhiteBishop",{3,7}}],Debug->True]//Timing

GraphData[{"WhiteBishop",{3,7}},"Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"WhiteBishop",{3,7}},"VertexList"],GraphData[{"WhiteBishop",{3,7}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"WhiteBishop",{3,7}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance exact construction

CenterGraph@RotateGraph[,{3,8}]

v=UnitCirclePoints[4,Pi/2];
v=Join[v,UnitDistancePoint[Sequence@@v[[#]]]&/@{{1,2},{4,1},{3,4},{2,3}}];
v=Join[v,UnitDistancePoint[Sequence@@v[[#]]]&/@{{7,6},{5,8}}];
g=Graph[Range@Length@v,UnitEdges[v],VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

UnitDistanceGraphEmbeddingQ@g

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,7}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,7}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

RotateGraph[,{6,4}]

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"WhiteBishop",{3,7}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"WhiteBishop",{3,7}}],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

g=CenterGraph@RotateGraph[,{11,26}]

gs=UnlabelGraph/@{,}

UnitDistanceGraphEmbeddingQ/@gs

Matchstick

GraphData[{"WhiteBishop",{3,7}},"Matchstick"]

GraphData[{"WhiteBishop",{3,7}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"WhiteBishop",{3,7}},"Edges"]]

3D All

GraphData[{"WhiteBishop",{3,7}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"WhiteBishop",{3,7}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"WhiteBishop",{3,7}},"Polyhedral"]

GraphData[{"WhiteBishop",{3,7}},"Graph",{"3D","Polyhedron"}]

Properties m×n

Properties n×n

Acyclic

Table[GraphData[{"WhiteBishop",{n,n}},"Acyclic"],{n,2,10}]

Arboricity

Possibly 1, 2, 2, 3, 4, 4, 5, 6, 6, ...

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

ArcTransitivity

Table[GraphData[{"WhiteBishop",{n,n}},"ArcTransitivity"],{n,2,10}]

ArcTransitive

Table[GraphData[{"WhiteBishop",{n,n}},"ArcTransitive"],{n,2,10}]

ArcTransitivity

Table[GraphData[{"WhiteBishop",{n,n}},"ArcTransitivity"],{n,2,10}]

AutomorphismCount

A158302

Table[GraphData[{"WhiteBishop",{n,n}},"AutomorphismCount"],{n,2,10}]

seq=Table[GroupOrder[GraphAutomorphismGroup[WhiteBishopGraph[n,n]]],{n,2,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{0,4},{

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

(2^(2Ceiling[Range[20]/2]-1))

Rest[%]==seq

BalabanIndex

Table[GraphData[{"WhiteBishop",{n,n}},"BalabanIndex"],{n,2,7}]

Bandwidth

Table[GraphData[{"WhiteBishop",{n,n}},"Bandwidth"],{n,2,10}]

BridgeCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"BridgeCount"],{n,2,10}]

BurningNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"BurningNumber"],{n,2,10}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"CharacteristicPolynomial"],{n,k0=2,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]

ChordlessCycleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ChordlessCycleCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

ChromaticInvariant

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ChromaticInvariant"],{n,2,10}]

FindSequenceFunction[Drop[seq,2],n]

ChromaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ChromaticNumber"],{n,2,10}]

ChromaticPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"ChromaticPolynomial"],{n,k0=2,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]

Circumference

A007590

seq=Table[GraphData[{"WhiteBishop",{n,n}},"Circumference"],{n,2,10}]

FindSequenceFunction[seq,n-1]

Table[Floor[n^2/2],{n,2,20}]

Classes

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

CliqueCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"CliqueCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

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

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

CliqueCoveringNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"CliqueCoveringNumber"],{n,2,10}]

FindSequenceFunction[Rest@seq,n-3]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

CliqueNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"CliqueNumber"],{n,2,10}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"CliquePolynomial"][x],{n,k0=2,kmax=10}])//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]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ComplementOddChordlessCycleCount"],{n,2,10}]

FindSequenceFunction[Take[seq,{3,5}],n]

ConnectedComponentCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedComponentCount"],{n,2,10}]

ConnectedDominatingSetCount

A289169

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedDominatingSetCount"],{n,2,10}]

FindSequenceFunction[DeleteMissing[seq],n-1]

ConnectedDominationNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedDominationNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]

ConnectedDominationPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedDominationPolynomial"],{n,k0=3,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

A290769

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedInducedSubgraphCount"],{n,2,10}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

ConnectedInducedSubgraphPolynomial

(poly=DeleteCases[Factor@Table[GraphData[{"WhiteBishop",{n,n}},"ConnectedInducedSubgraphPolynomial"][x],{n,k0=2,kmax=10}],_Missing[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[{"WhiteBishop",{n,n}},"CrossingNumber"],{n,2,10}]

CycleCount

A234630

seq=Table[GraphData[{"WhiteBishop",{n,n}},"CycleCount"],{n,2,10}]

seq={1,53,4943,12300529,134480655082,84649132742336262,212615053823090089357638};

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n-2]

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

CyclePolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"CyclePolynomial"][x],{n,k0=3,kmax=10}])//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]

Cyclic

Table[GraphData[{"WhiteBishop",{n,n}},"Cyclic"],{n,2,10}]

CyclomaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"CyclomaticNumber"],{n,2,10}]

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

Table[(1/18) (45-27 (2^n)+7 (3^n)),{n,2,10}]

DetourIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"DetourIndex"],{n,2,10}]

FindSequenceFunction[{1,35,550,5566,49726},n]

FindGeneratingFunction[{1,35,550,5566,49726},x]

DetourMatrix

MatrixForm/@Table[GraphData[{"WhiteBishop",{n,n}},"DetourMatrix"],{n,2,10}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,2,10}]

DetourPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"DetourPolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"Diameter"],{n,2,10}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"WhiteBishop",{n,n}},"DistanceMatrix"],{n,2,10}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,2,10}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"DistancePolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"DistinguishingNumber"],{n,2,10}]

DomaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"DomaticNumber"],{n,2,10}]

DominatingSetCount

A289170

seq=Table[GraphData[{"WhiteBishop",{n,n}},"DominatingSetCount"],{n,2,10}]

DominationNumber

A004524

seq=Table[GraphData[{"WhiteBishop",{n,n}},"DominationNumber"],{n,2,10}]

(seq=Monitor[Table[DominationNumber[WhiteBishopGraph[n,n]],{n,2,18}],n])//Timing

DominationPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"DominationPolynomial"],{n,k0=2,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]

EdgeChromaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"EdgeChromaticNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

EdgeConnectivity

Table[GraphData[{"WhiteBishop",{n,n}},"EdgeConnectivity"],{n,2,10}]

EdgeCount

A289179

seq=Table[GraphData[{"WhiteBishop",{n,n}},"EdgeCount"],{n,2,10}]

(seq=Monitor[Table[EdgeCount[WhiteBishopGraph[n,n]],{n,2,30}],n])//Timing

coef=FindLinearRecurrence[seq]

Take[seq,Length[coef]]

LinearRecurrence[{2,1,-4,1,2,-1},{0,1,4,14,28,55},20]

Table[(n-1) (4 (n^2)-2n-3+3(((-1))^n))/12,{n,50}]

ToString[(n-1) (4 (n^2)-2n-3+3(((-1))^n))/12,InputForm]

gf=x FindGeneratingFunction[seq,x]

ToString[gf,InputForm]

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

CoefficientList[Series[(((x+4 (x^2)+(x^3)+2 (x^4)))/((((-1+x))^4) (((1+x))^2))),{x,0,20}],x]

EdgeCoverCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"EdgeCoverCount"],{n,2,10}]

EdgeCoverNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"EdgeCoverNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]

x FindGeneratingFunction[seq,x]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"EdgeCoverPolynomial"][x],{n,k0=2,kmax=10}])//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]

EulerianCycleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"EulerianCycleCount"],{n,2,10}]

FaceCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"FaceCount"],{n,2,10}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"FlowPolynomial"][x],{n,k0=2,kmax=10}])//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]

FractionalChromaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"FractionalChromaticNumber"],{n,2,10}]

RSolve[{a[n]==a[n-1]+1/a[n-1],a[1]==1},a[n],n]

RecurrenceTable[{a[n]==a[n-1]+1/a[n-1],a[1]==1},a,{n,2,10}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"FractionalEdgeChromaticNumber"],{n,2,10}]

FindSequenceFunction[Drop[seq,3],n-3]

Table[3 2^(n-3)-1,{n,2,10}]

Girth

seq=Table[GraphData[{"WhiteBishop",{n,n}},"Girth"],{n,2,10}]

Gonality

seq=Table[GraphData[{"WhiteBishop",{n,n}},"Gonality"],{n,2,10}]

HamiltonConnected

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonConnected"],{n,2,10}]

HamiltonDecompositionCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonDecompositionCount"],{n,2,10}]

Hamiltonian

Table[GraphData[{"WhiteBishop",{n,n}},"Hamiltonian"],{n,2,10}]

HamiltonianCycleCount

1/2 * A282890

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonianCycleCount"],{n,2,10}]

HamiltonianNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonianNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]

gf=x FindGeneratingFunction[seq,x]

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

Table[-1+3 (2^(-2+n)),{n,2,10}]

HamiltonianPathCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonianPathCount"],{n,2,10}]

HamiltonianWalkCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HamiltonianWalkCount"],{n,2,10}]

HararyIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HararyIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FunctionExpand

FindGeneratingFunction[seq,x]

SeriesCoefficient[((2 x-5 (x^2)-5 x -(x^4))/(2 (1-10 x+35 (x^2)-50 x +24 (x^4))))x,{x,0,n}]//FullSimplify

SeriesCoefficient[((2 x-5 (x^2)-5 x -(x^4))/(2 (1-10 x+35 (x^2)-50 x +24 (x^4)))),{x,0,n}]//FullSimplify

Table[(3/4)-15 (2^(-4+n))+((7 (3^(-2+n)))/4)+9 (4^(-3+n)),{n,2,10}]

Table[(1/48)(36-45 (2^n)+28 (3^(n-1))+27 (4^(n-1))),{n,2,10}]

HexagonCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"HexagonCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindSequenceFunction[Drop[seq,3],n-3]

FindGeneratingFunction[seq,x]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"IdiosyncraticPolynomial"][x,y],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"IndependenceNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]

IndependencePolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"IndependencePolynomial"][x],{n,k0=2,kmax=10}])//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]

IndependenceRatio

seq=Table[GraphData[{"WhiteBishop",{n,n}},"IndependenceRatio"],{n,2,10}]

FindSequenceFunction[seq,n-1]

IndependentEdgeSetCount

A286423

seq=Table[GraphData[{"WhiteBishop",{n,n}},"IndependentEdgeSetCount"],{n,2,10}]

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-1]

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

IndependentVertexSetCount

A216078

seq=Table[GraphData[{"WhiteBishop",{n,n}},"IndependentVertexSetCount"],{n,2,10}]

seq={1,1,3,7,27,87,409,1657,9089,43833,272947,1515903,10515147,65766991,501178937,3473600465,28773452321,218310229201,1949230218691,16035686850327,153281759047387,1356791248984295,1380621506668543};

FindLinearRecurrence[Drop[seq,2]]

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

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

IntersectionNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"IntersectionNumber"],{n,2,10}]

IrredundantSetCount

A290712

Table[GraphData[{"WhiteBishop",{n,n}},"IrredundantSetCount"],{n,2,10}]

KirchhoffIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"KirchhoffIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]

KirchhoffSumIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"KirchhoffSumIndex"],{n,2,10}]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"WhiteBishop",{n,n}},"LaplacianMatrix"],{n,2,10}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"LaplacianPolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"LeafCount"],{n,2,10}]

LongestCycleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"LongestCycleCount"],{n,2,10}]

LovaszNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"LovaszNumber"],{n,2,10}]

MatchingGeneratingPolynomial

(poly=Table[GraphData[{"WhiteBishop",{n,n}},"MatchingGeneratingPolynomial"],{n,k0=3,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]

MatchingNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MatchingNumber"],{n,2,10}]

FullSimplify[FindSequenceFunction[seq,n-1],n∈Integers]

FullSimplify[%,Mod[n,2]==#]&/@{0,1}

MatchingPolynomial

(poly=Table[GraphData[{"WhiteBishop",{n,n}},"MatchingPolynomial"],{n,k0=3,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[{"WhiteBishop",{n,n}},"MaximalCliqueCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FunctionExpand

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

Table[(1/6) (3-9 (2^(n-1))+7 (3^(n-1))),{n,2,10}]

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximalIndependentEdgeSetCount"],{n,2,10}]

MaximalIndependentVertexSetCount

A290613

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximalIndependentVertexSetCount"],{n,2,10}]

MaximumCliqueCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximumCliqueCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FunctionExpand

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

Table[(1/6) (3-9 (2^(n-1))+7 (3^(n-1))),{n,2,10}]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximumIndependentEdgeSetCount"],{n,2,10}]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximumIndependentVertexSetCount"],{n,2,10}]

MaximumLeafNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximumLeafNumber"],{n,2,10}]

MaximumVertexDegree

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MaximumVertexDegree"],{n,2,10}]

MeanDistance

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MeanDistance"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindGeneratingFunction[seq,x]

MinimalDominatingSetCount

A287897

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimalDominatingSetCount"],{n,2,10}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimalDominatingSetSignature"],{n,2,10}]//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumCliqueCoveringCount"],{n,2,10}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumConnectedDominatingSetCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumDistinguishingLabelingCount"],{n,2,10}]

MinimumDominatingSetCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumDominatingSetCount"],{n,2,10}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumEdgeCoverCount"],{n,2,10}]

MinimumVertexDegree

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumVertexDegree"],{n,2,10}]

MolecularTopologicalIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MolecularTopologicalIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindGeneratingFunction[seq,x]//Factor

-36(3·(2^n)-8)(27·(2^n)-(143^n)-18)

Table[(3 2^n-8)(14 3^n7 2^n+18)/36,{n,2,10}]

Nonhamiltonian

Table[GraphData[{"WhiteBishop",{n,n}},"Nonhamiltonian"],{n,2,10}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"WhiteBishop",{n,n}},"NormalizedLaplacianMatrix"],{n,2,10}]

OddChordlessCycleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"OddChordlessCycleCount"],{n,2,10}]

PathCount

A291999

seq=Table[GraphData[{"WhiteBishop",{n,n}},"PathCount"],{n,2,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

PathPolynomial

(poly=DeleteMissing@Factor[Table[GraphData[{"WhiteBishop",{n,n}},"PathPolynomial"],{n,k0=2,kmax=10}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

PentagonCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"PentagonCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindSequenceFunction[Drop[seq,2],n]

FindGeneratingFunction[seq,x]

FindLinearRecurrence[seq]

Perfect

Table[GraphData[{"WhiteBishop",{n,n}},"Perfect"],{n,2,10}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ProjectivePlaneCrossingNumber"],{n,2,10}]

Radius

seq=Table[GraphData[{"WhiteBishop",{n,n}},"Radius"],{n,2,10}]

RankPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"RankPolynomial"][x,y],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"RectilinearCrossingNumber"],{n,2,10}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"ReliabilityPolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"SigmaPolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"SpanningTreeCount"],{n,2,10}]

FindSequenceFunction[Take[seq,7],n]

FindGeneratingFunction[Take[seq,7],x]

SquareCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"SquareCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindSequenceFunction[Drop[seq,3],n-3]

FindGeneratingFunction[seq,x]

FindLinearRecurrence[Drop[seq,3]]

StabilityIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"StabilityIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"TopologicalIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"ToroidalCrossingNumber"],{n,2,10}]

Traceable

Table[GraphData[{"WhiteBishop",{n,n}},"Traceable"],{n,2,10}]

TriangleCount

seq=Table[GraphData[{"WhiteBishop",{n,n}},"TriangleCount"],{n,2,10}]

TuttePolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"TuttePolynomial"][x,y],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"Untraceable"],{n,2,10}]

VertexConnectivity

Table[GraphData[{"WhiteBishop",{n,n}},"VertexConnectivity"],{n,2,10}]

VertexCount

A007590

seq=Table[GraphData[{"WhiteBishop",{n,n}},"VertexCount"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FullSimplify[#,n∈Integers]&

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

Table[Floor[n^2/2],{n,20}]

VertexCoverNumber

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"WhiteBishop",{n,n}},"VertexCoverNumber"],{n,2,10}]

FindSequenceFunction[seq,n-1]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

Table[3 (2^(-3+n)),{n,2,10}]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"WhiteBishop",{n,n}},"VertexCoverPolynomial"][x],{n,k0=2,kmax=10}])//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[{"WhiteBishop",{n,n}},"VertexDegrees"],{n,2,10}]

VertexTransitive

Table[GraphData[{"WhiteBishop",{n,n}},"VertexTransitive"],{n,2,10}]

WienerIndex

A292059

seq=Table[GraphData[{"WhiteBishop",{n,n}},"WienerIndex"],{n,2,10}]

(seq=Table[WienerIndex[WhiteBishopGraph[n,n]],{n,2,20}])//Timing

FindSequenceFunction[seq,n-1]

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,8]

seq=LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{1,8,42,104,251,464,852,1360},40]

FullSimplify[FindSequenceFunction[seq,n-1]]

Expand[(-1+2 (-1+n) n)]

Table[ (3+2 n-6 (n^2)-8 (n^3)+6 (n^4)+3 (((-1))^n) (-1-2 n+2 (n^2)))/24,{n,2,20}]

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

CoefficientList[Series[((-1-6 x-24 (x^2)-10 (x^3)-7 (x^4))/((((-1+x))^5) (((1+x))^3))),{x,0,20}],x]

WienerSumIndex

seq=Table[GraphData[{"WhiteBishop",{n,n}},"WienerSumIndex"],{n,2,10}]

FindSequenceFunction[seq,n-1]

GraphData

GraphDataClassString

GraphDataClassString["WhiteBishop"]

GraphDataString

Square

file=OpenWrite["white-bishop.m"]

Do[WriteString[file,GraphDataString[WhiteBishopGraph[n,n],
"Name"->"("<>ToString[n]<>","<>ToString[n]<>")\[Hyphen]white bishop graph",
"StandardName"->{"WhiteBishop",{n,n}},
"Classes"->{"WhiteBishop"},
"NotationRules"->{"WhiteBishop"->{n,n}},
"Information"->"WhiteBishopGraph",
"Skip"->{"ABCEnergy","ABCSpectralRadius","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","BetweennessCentralities","BurningNumber","CharacteristicPolynomial","Circumference","CliqueCoveringNumber","ConnectedDominatingSets","ConstructionNumber","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","Curvatures","CyclePolynomial","CyclicEdgeConnectivity","DetourMatrix","DominatingSets","EigenvectorCentralities","EmbeddingClasses","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","MinimumVertexCut","MinimumTotalDominatingSet","MinimumWeightFractionalEdgeColoring","MinimumWeightFractionalColoring","Minors","Outerplanar","PageRankCentralities","PathCoveringNumber","PathPolynomial","PathSignature","Paths","ProjectivePlanar","QuadraticEmbeddingConstant","RandicEnergy","RandicSpectralRadius","RectilinearCrossingCount","Skewness","SomborEnergy","SomborSpectralRadius","SpanningTreeCount","TopologicalIndex","TotalDominationPolynomial","TuttePolynomial","VertexCuts"},
Debug->True,
TimeConstraint->10
]],{n,11,20,2}]

Close[file]