Triangular Honeycomb Bishop Graph

Author

Eric W. Weisstein
December 17, 2024

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

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

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

Source

Wagon (2014)

Figure

3

g=TriangularHoneycombBishopGraph[3]

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

strightedges=Select[edges,EuclideanDistance@@v[[#]]<1.01&]

curvededges=Complement[edges,strightedges]

Append[#,.5]&/@curvededges

perp[p1_,p2_,x_,y_]:=Module[{u=p2-p1},p1+x u+y{-#2,#}&@@u]

g3=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,strightedges],
BezierCurve[{v[[#]],perp[v[[#]],v[[#2]],.5,#3/2],v[[#2]]}]&@@@{{1,4,-.5},{1,6,.5}},
FaceForm[White],Disk[v,.1]
},
PlotLabel->"3\[Hyphen]triangular honeycomb bishop graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

4

g=TriangularHoneycombBishopGraph[4]

EdgeLengths[g]//Tally

v=GraphEmbedding[g];

edges=List@@@EdgeList[g];

strightedges=Select[edges,EuclideanDistance@@v[[#]]<1.01&]

curvededges=Complement[edges,strightedges]

perp[p1_,p2_,x_,y_]:=Module[{u=p2-p1},p1+x u+y{-#2,#}&@@u]

curvededgesandoffsets={{1,4,-0.5},{1,6,0.5},{1,7,-1},{1,10,1},{2,7,-0.5},{2,9,0.5},{3,8,-0.5},{3,10,0.5}};
g4=Graphics[{
FaceForm[LightGray],EdgeForm[Black],Polygon[CirclePoints[#,{1/Sqrt[3],Pi/2},6]]&/@v,
Thick,Line[v,strightedges],
BezierCurve[{v[[#]],perp[v[[#]],v[[#2]],.5,#3/2],v[[#2]]}]&@@@curvededgesandoffsets,
FaceForm[White],Disk[v,.1]
},
PlotLabel->"4\[Hyphen]triangular honeycomb bishop graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

Combined

GraphicsRow[{g3,g4},ImageSize->600,Alignment->Top]

Special cases

RecognizeGraph/@Table[TriangularHoneycombBishopGraph[n],{n,20}]

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

Classes

GraphClassesThatAre["TriangularHoneycombBishop"]

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

StringReplace[StringRiffle[GraphData[#,"Description"]&/@%,", "]," graph"->""]

Layouts

Graphs.m

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

Curved (doubled) edges

GraphicsRow[StyleGraphs@Table[TriangularHoneycombBishopGraph[n]/.g_Graph:>AdjacencyGraph[AdjacencyMatrix[g],EdgeShapeFunction->"Line",DirectedEdges->True,VertexCoordinates->GraphEmbedding[g]],{n,5}],Dividers->All,ImageSize->500]

Edges

GraphicsRow[StyleGraphs@Table[Graph@@Through[{VertexList,EdgeList}[TriangularHoneycombBishopGraph[n]]],{n,5}],Dividers->All,ImageSize->500]

GraphData default

Table[GraphData[{"TriangularHoneycombBishop",n},"#"],{n,6}]

GraphicsRow[StyleGraphs@Table[GraphData[{"TriangularHoneycombBishop",n}],{n,6}],Dividers->All,ImageSize->500]

GraphData "TriangularGrid"

GraphicsRow[StyleGraphs@Table[GraphData[{"TriangularHoneycombBishop",n},"Graph","TriangularGrid"][[1]],{n,6}],Dividers->All,ImageSize->500]

Table[GraphData[{"TriangularHoneycombBishop",n},"Graph","TriangularGrid"][[1]],{n,20}]

GraphData non-degenerate

GraphicsRow[Table[StyleGraphs[
Pick[
GraphData[{"TriangularHoneycombBishop",n},"Graph","All"],
FreeQ[#,"Degenerate"]&/@GraphData[{"TriangularHoneycombBishop",n},"EmbeddingClasses"]
][[1]]],{n,6}],Dividers->All,ImageSize->500]

Embeddings

<<MathWorld`Graphs`

{"TriangularHoneycombBishop", 1} == "SingletonGraph"

Names

GraphData[{"TriangularHoneycombBishop",1},"StandardName"]

GraphData[{"TriangularHoneycombBishop",1},"StandardNames"]

{"TriangularHoneycombBishop", 2} == {"Path", 3}

Names

GraphData[{"TriangularHoneycombBishop",2},"StandardName"]

GraphData[{"TriangularHoneycombBishop",2},"StandardNames"]

TriangularGrid

GraphData[{"TriangularHoneycombBishop",2},"Graph","TriangularGrid"]

{"TriangularHoneycombBishop", 3} == "CisSquareWithTwoTriangles"

Names

GraphData[{"TriangularHoneycombBishop",3},"StandardName"]

GraphData[{"TriangularHoneycombBishop",3},"StandardNames"]

{"TriangularHoneycombBishop", 4} == {"BlackBishop", {4, 5}}

Names

GraphData[{"TriangularHoneycombBishop",4},"StandardName"]

GraphData[{"TriangularHoneycombBishop",4},"StandardNames"]

{"TriangularHoneycombBishop", 5} == {"BlackBishop", {5, 6}}

Names

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

GraphData[{"TriangularHoneycombBishop",5},"StandardNames"]

{"TriangularHoneycombBishop", 6} == {"BlackBishop", {6, 7}}

Names

GraphData[{"TriangularHoneycombBishop",6},"StandardName"]

GraphData[{"TriangularHoneycombBishop",6},"StandardNames"]

{"TriangularHoneycombBishop", 7} == {"BlackBishop", {7, 8}}

Names

GraphData[{"TriangularHoneycombBishop",7},"StandardName"]

GraphData[{"TriangularHoneycombBishop",7},"StandardNames"]

{"TriangularHoneycombBishop", 8} == {"BlackBishop", {8, 9}}

Names

GraphData[{"TriangularHoneycombBishop",8},"StandardName"]

GraphData[{"TriangularHoneycombBishop",8},"StandardNames"]

{"TriangularHoneycombBishop", 9} == {"BlackBishop", {9, 10}}

Names

GraphData[{"TriangularHoneycombBishop",9},"StandardName"]

GraphData[{"TriangularHoneycombBishop",9},"StandardNames"]

{"TriangularHoneycombBishop", 10} == {"BlackBishop", {10, 11}}

Names

GraphData[{"TriangularHoneycombBishop",10},"StandardName"]

GraphData[{"TriangularHoneycombBishop",10},"StandardNames"]

{"TriangularHoneycombBishop", 11} == {"BlackBishop", {11, 12}}

Names

GraphData[{"TriangularHoneycombBishop",11},"StandardName"]

GraphData[{"TriangularHoneycombBishop",11},"StandardNames"]

{"TriangularHoneycombBishop", 12} == {"BlackBishop", {12, 13}}

Names

GraphData[{"TriangularHoneycombBishop",12},"StandardName"]

GraphData[{"TriangularHoneycombBishop",12},"StandardNames"]

{"TriangularHoneycombBishop", 13} == {"BlackBishop", {13, 14}}

Names

GraphData[{"TriangularHoneycombBishop",13},"StandardName"]

GraphData[{"TriangularHoneycombBishop",13},"StandardNames"]

{"TriangularHoneycombBishop", 14} == {"BlackBishop", {14, 15}}

Names

GraphData[{"TriangularHoneycombBishop",14},"StandardName"]

GraphData[{"TriangularHoneycombBishop",14},"StandardNames"]

{"TriangularHoneycombBishop", 15} == {"BlackBishop", {15, 16}}

Names

GraphData[{"TriangularHoneycombBishop",15},"StandardName"]

GraphData[{"TriangularHoneycombBishop",15},"StandardNames"]

{"TriangularHoneycombBishop", 16} == {"BlackBishop", {16, 17}}

Names

GraphData[{"TriangularHoneycombBishop",16},"StandardName"]

GraphData[{"TriangularHoneycombBishop",16},"StandardNames"]

{"TriangularHoneycombBishop", 17} == {"BlackBishop", {17, 18}}

Names

GraphData[{"TriangularHoneycombBishop",17},"StandardName"]

GraphData[{"TriangularHoneycombBishop",17},"StandardNames"]

{"TriangularHoneycombBishop", 18} == {"BlackBishop", {18, 19}}

Names

GraphData[{"TriangularHoneycombBishop",18},"StandardName"]

GraphData[{"TriangularHoneycombBishop",18},"StandardNames"]

{"TriangularHoneycombBishop", 19} == {"BlackBishop", {19, 20}}

Names

GraphData[{"TriangularHoneycombBishop",19},"StandardName"]

GraphData[{"TriangularHoneycombBishop",19},"StandardNames"]

{"TriangularHoneycombBishop", 20} == {"BlackBishop", {20, 21}}

Names

GraphData[{"TriangularHoneycombBishop",20},"StandardName"]

GraphData[{"TriangularHoneycombBishop",20},"StandardNames"]

Properties

Acyclic

Table[GraphData[{"TriangularHoneycombBishop",n},"Acyclic"],{n,9}]

AdjacencyMatrixCount

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

Table[((Binomial[n+1,2]!)/(Piecewise[{{1,n==1},{6,n==2},{432,n==4},{108,Mod[n,3]==1}},48])),{n,10}]

Arboricity

Table[GraphData[{"TriangularHoneycombBishop",n},"Arboricity"],{n,9}]

ArcTransitive

Table[GraphData[{"TriangularHoneycombBishop",n},"ArcTransitive"],{n,9}]

ArcTransitivity

Table[GraphData[{"TriangularHoneycombBishop",n},"ArcTransitivity"],{n,9}]

AutomorphismCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"AutomorphismCount"],{n,9}]

%==seq

BalabanIndex

Table[GraphData[{"TriangularHoneycombBishop",n},"BalabanIndex"],{n,9}]

Bandwidth

Table[GraphData[{"TriangularHoneycombBishop",n},"Bandwidth"],{n,9}]

BridgeCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"BridgeCount"],{n,9}]

BurningNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"BurningNumber"],{n,9}]

CharacteristicPolynomial

(poly=Table[GraphData[{"TriangularHoneycombBishop",n},"CharacteristicPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

ChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ChordlessCycleCount"],{n,9}]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[Drop[DeleteMissing@seq,6],n]

ChromaticInvariant

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ChromaticInvariant"],{n,9}]

ChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ChromaticNumber"],{n,9}]

ChromaticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"ChromaticPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Circumference"],{n,9}]

FindLinearRecurrence[Drop[seq,3]]

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

FindSequenceFunction[Drop[seq,4],n-4]

Classes

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

CliqueCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CliqueCount"],{n,9}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

CliqueCoveringNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CliqueCoveringNumber"],{n,9}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

CliqueNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CliqueNumber"],{n,9}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"CliquePolynomial"][x],{n,k0=1,kmax=20}])//Column

FindSequenceFunction[#,n-3]&/@Transpose[Drop[CoefficientList[#,x]&/@poly,3]]//Factor

FullSimplify[1+(1/2) n (1+n)x+(3/2) (-2+n) (-1+n)(x^2)+(-3+n) (-2+n)(x^3)]

Table[Piecewise[{{1+x,n==1}},1+(1/2) n (1+n)x+(3/2) (-2+n) (-1+n)(x^2)+(-3+n) (-2+n)(x^3)],{n,9}]//Column

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ComplementOddChordlessCycleCount"],{n,9}]

ConnectedComponentCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ConnectedComponentCount"],{n,9}]

ConnectedComponents

GraphData[{"TriangularHoneycombBishop",4},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombBishop",5},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombBishop",6},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombBishop",7},"ConnectedComponents","Name"]

GraphData[{"TriangularHoneycombBishop",8},"ConnectedComponents","Name"]

ConnectedDominationNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ConnectedDominationNumber"],{n,9}]

FindSequenceFunction[seq,n]

ConnectedInducedSubgraphCount

A290783

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ConnectedInducedSubgraphCount"],{n,9}]

FindLinearRecurrence[DeleteMissing@seq]

CrossingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CrossingNumber"],{n,9}]

CycleCount

A290782

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CycleCount"],{n,9}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing[seq],n]

CyclePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"CyclePolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"TriangularHoneycombBishop",n},"Cyclic"],{n,9}]

CyclomaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"CyclomaticNumber"],{n,9}]

FindSequenceFunction[seq,n]

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

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

DetourIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DetourIndex"],{n,9}]

DetourMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombBishop",n},"DetourMatrix"],{n,5}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,5}]

DetourPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"DetourPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

Diameter

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Diameter"],{n,9}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombBishop",n},"DistanceMatrix"],{n,5}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,5}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"DistancePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

DistinguishingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DistinguishingNumber"],{n,9}]

DomaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DomaticNumber"],{n,9}]

DominatingSetCount

A290941

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DominatingSetCount"],{n,9}]

FindLinearRecurrence[seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

DominationNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"DominationNumber"],{n,9}]

(seq=Monitor[Table[DominationNumber[TriangularHoneycombBishopGraph[n]],{n,30}],n])//Timing

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

DominationPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"DominationPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeChromaticNumber"],{n,9}]

EdgeConnectivity

Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeConnectivity"],{n,9}]

EdgeCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeCount"],{n,9}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]//Factor

EdgeCoverCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeCoverCount"],{n,9}]

FindSequenceFunction[DeleteMissing@seq,n]

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

EdgeCoverNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeCoverNumber"],{n,9}]

(seq=Monitor[Table[EdgeCoverNumber[TriangularHoneycombBishopGraph[n]],{n,9}],n])//Timing

FindSequenceFunction[seq,n]

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

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"EdgeCoverPolynomial"],{n,k0=1,kmax=20}]/.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,9}]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"EulerianCycleCount"],{n,9}]

FaceCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"FaceCount"],{n,9}]

FindSequenceFunction[seq,n]

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

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

FlowPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"FlowPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])

FractionalChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"FractionalChromaticNumber"],{n,9}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"FractionalEdgeChromaticNumber"],{n,9}]

Girth

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Girth"],{n,9}]

Gonality

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Gonality"],{n,9}]

HamiltonConnected

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonConnected"],{n,9}]

HamiltonDecompositionCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonDecompositionCount"],{n,9}]

Hamiltonian

Table[GraphData[{"TriangularHoneycombBishop",n},"Hamiltonian"],{n,9}]

HamiltonianCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonianCycleCount"],{n,9}]

HamiltonianNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonianNumber"],{n,9}]

HamiltonianPathCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonianPathCount"],{n,9}]

HamiltonianWalkCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HamiltonianWalkCount"],{n,9}]

HararyIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HararyIndex"],{n,9}]

(seq=Monitor[Table[HararyIndex[TriangularHoneycombBishopGraph[n]],{n,9}],n])//Timing

FindSequenceFunction[seq,n]

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

HexagonCount

A290779

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"HexagonCount"],{n,9}]

(seq=Table[CycleCount[TriangularHoneycombBishopGraph[n],6],{n,20}])//Timing

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,0,1,57,486,2360,8394,24354},40]

FindSequenceFunction[seq,n]//Factor

Binomial[n+1,4]

24/1680

Factor[(-62+11 n-109 (n^2)+40 (n^3))]

Table[Binomial[n+1,4] (-62+11 n-109 (n^2)+40 (n^3))/70,{n,20}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[(((x^2)+49 (x^3)+58 (x^4)+12 (x^5))/(((-1+x))^8)),{x,0,20}],x]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"IdiosyncraticPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//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,9}]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IndependenceNumber"],{n,9}]

FindLinearRecurrence[seq]

LinearRecurrence[{1,0,2,-2,0,-1,1},{1,3,3,5,9,9,12},50]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,50}]

seq=Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,1,50,3}]

FindSequenceFunction[%,(n-1)/3+1]//Simplify

Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,2,50,3}]

FindSequenceFunction[%,(n-2)/3+1]//FullSimplify

seq=Table[(1/18) ((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]),{n,3,50,3}]

FindSequenceFunction[seq,n/3]//FullSimplify

Table[Piecewise[{{n(n+3),Mod[n,3]==0},{(n+1) (n+2),Mod[n,3]==1},{(n+1) (n+4),Mod[n,3]==2}}]/6,{n,50}]

IndependencePolynomial

(poly=Table[GraphData[{"TriangularHoneycombBishop",n},"IndependencePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x])//Column

Exponent[#,x]&/@DeleteMissing[poly]

CoefficientList[Take[poly,10],x]//Column

poly=Import["https://oeis.org/A244500/b244500.txt","Table"][[All,2]]

coef=FindLinearRecurrence[DeleteMissing[poly]]

coef=FindLinearRecurrence[Drop[DeleteMissing[poly],2]]

coef=FindPolynomialRecurrence[DeleteMissing[poly],6]

coef=FindPolynomialRecurrence[DeleteMissing@Drop[poly,2]

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,9}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IndependenceRatio"],{n,9}]

Table[(1/18) ((((n+3) (3n+2)-2 (n+3) Cos[((2 n π)/3)]-2 (Sqrt[3]) (n+1) Sin[((2 n π)/3)]))/(Binomial[n+1,2])),{n,9}]

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A293249

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IndependentEdgeSetCount"],{n,9}]

seq={1,3,24,624,58258,21423356,33352210384,234271975272172,7849205797114197704,1318509837007765330964948,1161589241831987552884873549832,5592410686612429649991093735982825800,152814054664056604403587836700924195641924920};

Head/@Table[FindLinearRecurrence[Drop[seq,k]],{k,0,3}]

FindGeneratingFunction[seq,x]//Head//Timing

IndependentVertexSetCount

A000110

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IndependentVertexSetCount"],{n,9}]

Table[BellB[n+1],{n,9}]

IntersectionNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"IntersectionNumber"],{n,9}]

KirchhoffIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"KirchhoffIndex"],{n,9}]

FindSequenceFunction[seq,n]

KirchhoffSumIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"KirchhoffSumIndex"],{n,9}]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombBishop",n},"LaplacianMatrix"],{n,9}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"LaplacianPolynomial"],{n,k0=1,kmax=20}]/.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,9}]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"LeafCount"],{n,9}]

LongestCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"LongestCycleCount"],{n,9}]

FindSequenceFunction[DeleteMissing@Drop[seq,6],n-6]

FindSequenceFunction[DeleteMissing@Drop[seq,7],n-7]

LovaszNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"LovaszNumber"],{n,9}]

MatchingGeneratingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"MatchingGeneratingPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MatchingNumber"],{n,9}]

Do[Print[{n,MatchingNumber[TriangularHoneycombBishopGraph[n]]//Timing}],{n,40}]

(seq=Monitor[Table[MatchingNumber[TriangularHoneycombBishopGraph[n]],{n,50}],n])//Timing

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,3]]

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,13],n-13]//FullSimplify

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

Clear[a]

Do[a[i]=seq[[i+2]],{i,11}]

-36-11 \[FormalN]-(\[FormalN]^2)+2 \[FormalY][\[FormalN]]+2 \[FormalY][6+\[FormalN]]==0/.\[FormalN]->-2+n/.n->n-4//Simplify

a[n_]:=a[n]=(n^2-n+6-2a[n-6])/2

seq=Array[a,70]

Table[a[i]==seq[[i]],{i,13}]

RecurrenceTable[{a[n]==(n^2-n+6-2a[n-6])/2,a[1]==0,a[2]==0,a[3]==3,a[4]==3,a[5]==6,a[6]==9,a[7]==12,a[8]==18,a[9]==21,a[10]==27,a[11]==33,a[12]==39,a[13]==45},a,{n,50}]

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

gf=(x^14) FindGeneratingFunction[Drop[seq,11],x]

Series[gf,{x,0,20}]

gf1=Sum[seq[[i]](x^(i+2)),{i,11}]+gf

Factor[gf1]

Series[gf1,{x,0,20}]

FullSimplify[SeriesCoefficient[gf1,{x,0,n}],n∈Integers&&n>0]

FullSimplify[Table[%50,{n,72}]-Join[{0,0},seq]]

Join[{0,0},seq]

seq2=Table[Piecewise[{{3,n==4},{12,n==7}},(1/12) (3 (-3+n+(n^2))+5 Cos[((n π)/2)]+4 (((-1))^n) (Cos[((n π)/3)]+(Sqrt[3]) Sin[((n π)/3)]) (Cos[((n π)/2)]-Sin[((n π)/2)])-5 Sin[((n π)/2)])],{n,72}]

seq2-Join[{0,0},seq]//Union

Table[Piecewise[{{3,n==4},{12,n==7}},(3 (n^2+n-3)+5 Cos[n Pi/2]+4 (-1)^n (Cos[n Pi/3]+Sqrt[3] Sin[n Pi/3]) (Cos[n Pi/2]-Sin[n Pi/2])-5 Sin[n Pi/2])/12],{n,50}]

Series[x^3*(3-6*x+6*x^2-3*x^3+3*x^5-3*x^6+3*x^8-3*x^9+3*x^11-3*x^12+x^13)/((1-x)^3*(1+x^2)*(1-x^2+x^4)),{x,0,30}]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"MatchingPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximalCliqueCount"],{n,9}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>1]//Factor

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximalIndependentEdgeSetCount"],{n,9}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximalIndependentVertexSetCount

A290615

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximalIndependentVertexSetCount"],{n,9}]

seq=Import["https://oeis.org/A290615/b290615.txt","Table"][[All,2]]

FindLinearRecurrence[DeleteMissing@seq]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]//Head//Timing

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximalIndependentVertexSetSignature

A290724

Table[GraphData[{"TriangularHoneycombBishop",n},"MaximalIndependentVertexSetSignature"],{n,9}]//Column

MaximumCliqueCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximumCliqueCount"],{n,9}]

FindSequenceFunction[seq,n]

gf=FindGeneratingFunction[seq,x]//Factor

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

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximumIndependentEdgeSetCount"],{n,9}]

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[DeleteMissing@Drop[seq,3]]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximumIndependentVertexSetCount"],{n,9}]

MaximumLeafNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MaximumLeafNumber"],{n,9}]

MeanDistance

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MeanDistance"],{n,9}]

MinimalDominatingSetCount

A290875

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimalDominatingSetCount"],{n,9}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimalDominatingSetSignature

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimalDominatingSetSignature"],{n,9}]//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumCliqueCoveringCount"],{n,9}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumDistinguishingLabelingCount"],{n,9}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimumDominatingSetCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumDominatingSetCount"],{n,9}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumEdgeCoverCount"],{n,9}]

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

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

MolecularTopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MolecularTopologicalIndex"],{n,9}]

Nonhamiltonian

Table[GraphData[{"TriangularHoneycombBishop",n},"Nonhamiltonian"],{n,9}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombBishop",n},"NormalizedLaplacianMatrix"],{n,4}]

OddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"OddChordlessCycleCount"],{n,9}]

PathCount

A290774

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"PathCount"],{n,9}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=(x^3) FindGeneratingFunction[DeleteCases[seq,0|_Missing],x]

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

PathPolynomial

(poly=Factor[Table[GraphData[{"TriangularHoneycombBishop",n},"PathPolynomial"],{n,k0=1,kmax=9}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteCases[poly,0|_Missing]]

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

A290775

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"PentagonCount"],{n,9}]

(seq=Table[CycleCount[TriangularHoneycombBishopGraph[n],5],{n,20}])//Timing

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,2,24,138,532,1596},40]

FindSequenceFunction[seq,n]//Factor

Binomial[n+1,4]

Table[2/5Binomial[n+1,4](8-7 n+2 (n^2)),{n,20}]

gf=x FindGeneratingFunction[seq,x]

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

Perfect

Table[GraphData[{"TriangularHoneycombBishop",n},"Perfect"],{n,9}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ProjectivePlaneCrossingNumber"],{n,9}]

Radius

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Radius"],{n,9}]

RankPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"RankPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//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,9}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"RectilinearCrossingNumber"],{n,9}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"ReliabilityPolynomial"],{n,k0=1,kmax=20}]/.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,9}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"SigmaPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

SpanningTreeCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"SpanningTreeCount"],{n,9}]

SquareCount

A051843

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"SquareCount"],{n,9}]

(seq=Table[CycleCount[TriangularHoneycombBishopGraph[n],4],{n,15}])//Timing

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]//Factor

Table[(6n-1)Binomial[n+3,4]/5,{n,0,20}]

StabilityIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"StabilityIndex"],{n,9}]

FindLinearRecurrence[Drop[seq,2]]

FindSequenceFunction[seq,n]

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

gf=x FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"TopologicalIndex"],{n,9}]

ToroidalCrossingNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"ToroidalCrossingNumber"],{n,9}]

Traceable

Table[GraphData[{"TriangularHoneycombBishop",n},"Traceable"],{n,9}]

TriangleCount

A034827

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"TriangleCount"],{n,9}]

FindLinearRecurrence[Drop[seq,2]]

Table[2Binomial[n+1,4],{n,9}]

2Binomial[Range[0,20],4]

TuttePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"TuttePolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x,y])//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,9}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"TriangularHoneycombBishop",n},"Untraceable"],{n,9}]

VertexConnectivity

Table[GraphData[{"TriangularHoneycombBishop",n},"VertexConnectivity"],{n,9}]

VertexCount

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"VertexCount"],{n,20}]

FindSequenceFunction[seq,n]

Binomial[n+1,2]

VertexCoverNumber

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"VertexCoverNumber"],{n,9}]

Table[(1/2) n (1+n)-(1/18) ((3+n) (2+3 n)-2 (3+n) Cos[((2 n π)/3)]-2 (Sqrt[3]) (1+n) Sin[((2 n π)/3)]),{n,9}]

FullSimplify[(1/2) n (1+n)-(1/18) ((3+n) (2+3 n)-2 (3+n) Cos[((2 n π)/3)]-2 (Sqrt[3]) (1+n) Sin[((2 n π)/3)])]

FindLinearRecurrence[seq]

seq=LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,3,5,6,12,16},40]

FullSimplify[RSolve[{a[n]==a[n-1]+2a[n-3]-2a[n-4]-a[n-6]+a[n-7],a[1]==0,a[2]==0,a[3]==3,a[4]==5,a[5]==6,a[6]==12,a[7]==16},a[n],n],n∈Integers&&n>0]

Table[(1/(9 (123 I+47 (Sqrt[3]))))((E^((2 I n π)/3)) (114 I+132 (Sqrt[3])-9 I n+85 (Sqrt[3]) n)+(E^(-(2/3) I n π)) (255 I+9 (Sqrt[3])+(132 I-38 (Sqrt[3])) n)+(123 I+47 (Sqrt[3])) (-3+n (-1+3 n))),{n,10}]//N//Chop

FullSimplify[(1/(9 (123 I+47 (Sqrt[3]))))((E^((2 I n π)/3)) (114 I+132 (Sqrt[3])-9 I n+85 (Sqrt[3]) n)+(E^(-(2/3) I n π)) (255 I+9 (Sqrt[3])+(132 I-38 (Sqrt[3])) n)+(123 I+47 (Sqrt[3])) (-3+n (-1+3 n)))]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombBishop",n},"VertexCoverPolynomial"][x],{n,k0=1,kmax=20}])//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,9}]

FullSimplify[poly2-poly]

VertexDegrees

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

WienerIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"WienerIndex"],{n,9}]

FindSequenceFunction[seq,n]

WienerSumIndex

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"WienerSumIndex"],{n,9}]

GraphData

GraphDataClassString

GraphDataClassString["TriangularHoneycombBishop"]

GraphDataString

file=OpenAppend["~/trig-honeycomb-bishop.m"]

Do[WriteString[file,GraphDataString[BlackBishopGraph[n,n+1],
"Name"->"("<>ToString[n]<>","<>ToString[n+1]<>")\[Hyphen]black bishop graph",
"AlternateNames"->{ToString[n]<>"\[Hyphen]triangular honeycomb bishop graph"},
"StandardName"->{"BlackBishop",{n,n+1}},
"AlternateStandardNames"->{{"TriangularHoneycombBishop",n}},
"Classes"->{"BlackBishop","TriangularHoneycombBishop"},
"NotationRules"->{"BlackBishop"->{n,n+1},"TriangularHoneycombBishop"->n},
"Information"->"BlackBishopGraph",
"Skip"->{"ABCEnergy","ABCSpectralRadius","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","BurningNumber","ConnectedDominatingSet","ConnectedInducedSubgraphPolynomial","ConstructionNumber","CoolingNumber","Cycles","CyclicEdgeConnectivity","EdgeBetweennessCentralities","EdgeChromaticNumber","EigenvectorCentralities","EmbeddingClasses","HadwigerNumber","HITSCentralities","IdiosyncraticPolynomial","IndependenceNumber","IndependencePolynomial","IntersectionNumber","IrredundancePolynomial","KatzCentralities","LongestCycles","MatchingGeneratingPolynomial","MatchingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSets","MaximumIndependentEdgeSetCount","MaximumIndependentEdgeSets","MaximumIndependentVertexSet","MaximumIndependentVertexSetCount","MaximumIndependentVertexSets","MinimalPlanarIntegral","MinimumConnectedDominatingSet","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","MinimumEdgeColoring","MinimumTotalDominatingSet","MinimumVertexCover","MinimumVertexCut","Minors","PageRankCentralities","RandicEnergy","SomborEnergy","SomborSpectralRadius","TotalDominationPolynomial","TuttePolynomial"},
Debug->True,
TimeConstraint->10
]],{n,20,20}]

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