Triangular Honeycomb Rook

Author

Eric W. Weisstein
December 17, 2024

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

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

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

Figure

3

g=TriangularHoneycombRookGraph[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]]}]&@@@{{4,6,-0.5}},
FaceForm[White],Disk[v,.1]
},
PlotLabel->"3\[Hyphen]triangular honeycomb rook graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

4

g=TriangularHoneycombRookGraph[4]

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]

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]]}]&@@@{{4,6,-0.5},{7,9,-0.5},{7,10,-0.5},{8,10,-0.5}},
FaceForm[White],Disk[v,.1]
},
PlotLabel->"4\[Hyphen]triangular honeycomb rook graph",
LabelStyle->Directive[Black,FontFamily->"Times",14,Italic]]

Combined

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

Layouts

Graphs.m

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

Curved (doubled) edges

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

GraphData default

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

GraphData "TriangularGrid"

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

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

Classes

GraphClassesThatAre["TriangularHoneycombRook"]

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

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

Special cases

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

Properties

<<MathWorld`Graphs`

Acyclic

Table[GraphData[{"TriangularHoneycombRook",n},"Acyclic"],{n,20}]

AdjacencyMatrixCount

A022915

seq=Table[GraphData[{"TriangularHoneycombRook",n},"AdjacencyMatrixCount"],{n,20}]

Table[((Binomial[n+1,2]!)/(BarnesG[n+2])),{n,20}]

Table[Multinomial@@Range[n],{n,0,10}]

Table[((Binomial[n+1,2]!)/(BarnesG[n+2])),{n,20}]

FullSimplify[((Binomial[n+1,2]!)/(BarnesG[n+2]))]

Arboricity

Table[GraphData[{"TriangularHoneycombRook",n},"Arboricity"],{n,20}]

ArcTransitive

Table[GraphData[{"TriangularHoneycombRook",n},"ArcTransitive"],{n,20}]

ArcTransitivity

Table[GraphData[{"TriangularHoneycombRook",n},"ArcTransitivity"],{n,20}]

AutomorphismCount

A000178

seq=Table[GraphData[{"TriangularHoneycombRook",n},"AutomorphismCount"],{n,20}]

FindSequenceFunction[seq,n]

Table[BarnesG[n+2],{n,20}]

BarnesG[Range[2,10]]

BalabanIndex

Table[GraphData[{"TriangularHoneycombRook",n},"BalabanIndex"],{n,20}]

Bandwidth

Table[GraphData[{"TriangularHoneycombRook",n},"Bandwidth"],{n,20}]

BridgeCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"BridgeCount"],{n,20}]

BurningNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"BurningNumber"],{n,20}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"CharacteristicPolynomial"][x],{n,k0=1,kmax=10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

Product[x-k,{k,n-1}]

%//FunctionExpand

poly2=Table[((Gamma[x])/(Gamma[x-n+1]))x(((x+1))^(Binomial[n,2])),{n,20}]//FunctionExpand

FullSimplify[poly2/poly]

ChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ChordlessCycleCount"],{n,20}]

FindSequenceFunction[seq,n]

ChromaticInvariant

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ChromaticInvariant"],{n,20}]

ChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ChromaticNumber"],{n,20}]

ChromaticPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"ChromaticPolynomial"][x],{n,k0=1,kmax=10}])//Column

CoefficientList[Take[poly,5],x]

Table[Product[(((x-k))^(n-k)),{k,0,n-1}],{n,20}]//Column

Product[(((x-k))^(n-k)),{k,0,n-1}]//FullSimplify

Table[(E^((1/2) I (-1+n) n π+Derivative[1,0][Zeta][-1,1-x]-Derivative[1,0][Zeta][-1,n-x]-(n-x) (Derivative[1,0][Zeta][0,1-x]-Derivative[1,0][Zeta][0,n-x]))) (x^n),{n,2}]//FullSimplify

Table[Product[FactorialPower[x,k],{k,n}]//FunctionExpand,{n,20}]//Column

FullSimplify[Product[FactorialPower[x,k],{k,n}]]

Table[(I^(n (3+n))) (x^n) ((BarnesG[1-x])^(-1+n)) ((BarnesG[2-x])^(-n)) BarnesG[1+n-x],{n,4}]//FunctionExpand

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,20}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Circumference"],{n,20}]

Classes

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

CliqueCount

A000325

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CliqueCount"],{n,20}]

FindSequenceFunction[seq,n]

Table[2^n-n,{n,0,39}]

FindLinearRecurrence[%]

LinearRecurrence[{4,-5,2},{1,2,5},{0,20}]

CliqueCoveringNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CliqueCoveringNumber"],{n,20}]

CliqueNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CliqueNumber"],{n,20}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"CliquePolynomial"][x],{n,k0=1,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]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ComplementOddChordlessCycleCount"],{n,20}]

ConnectedComponentCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ConnectedComponentCount"],{n,20}]

ConnectedDominationNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ConnectedDominationNumber"],{n,20}]

ConnectedInducedSubgraphCount

A000295

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ConnectedInducedSubgraphCount"],{n,20}]

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

LinearRecurrence[{4,-5,2},{1,4,11},20]

CrossingNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CrossingNumber"],{n,20}]

Entity["Graph",{"Complete",k}]["CrossingNumber"]

Table[Sum[(1/4) Floor[(1/2) (-3+k)] Floor[(1/2) (-2+k)] Floor[(1/2) (-1+k)] Floor[(k/2)],{k,n}],{n,20}]

CycleCount

A289896

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CycleCount"],{n,20}]

Entity["Graph",{"Complete",k}]["CycleCount"]

sum=Entity["Graph",{"Complete",k}]["CycleCount"]//Activate

Table[sum,{k,10}]

seq=Table[Sum[sum,{k,n}],{n,20}]

FindLinearRecurrence[seq]

FindLinearRecurrence[Drop[seq,2]]

FindSequenceFunction[seq,n]

FindSequenceFunction[Drop[seq,2],n]

gf=x FindGeneratingFunction[seq,x]

Table[Sum[k(2HypergeometricPFQ[{1,1,1-k},{2},-1]-k-1)/4,{k,n}],{n,20}]

(1/2) (InactiveSum[Binomial[k,\[FormalK]] (-1+\[FormalK])!, \[FormalK], 3, k])

Table[Sum[Binomial[k,l](l-1)!,{k,n},{l,3,k}]/2,{n,30}]

CyclePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"CyclePolynomial"][x],{n,k0=1,kmax=10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindLinearRecurrence[Drop[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,20}]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"TriangularHoneycombRook",n},"Cyclic"],{n,20}]

CyclomaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"CyclomaticNumber"],{n,20}]

FindSequenceFunction[seq,n]

DetourIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DetourIndex"],{n,20}]

DetourMatrix

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

DetourPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"DetourPolynomial"][x],{n,k0=1,kmax=10}])//Column

Diameter

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Diameter"],{n,20}]

DistanceMatrix

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

DistancePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"DistancePolynomial"][x],{n,k0=1,kmax=10}])//Column

DistinguishingNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DistinguishingNumber"],{n,20}]

DomaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DomaticNumber"],{n,20}]

DominatingSetCount

A005329

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DominatingSetCount"],{n,20}]

FindSequenceFunction[seq,n]

FunctionExpand[%]

FullSimplify[(((-1))^n) QPochhammer[2,2,n]]

FullSimplify[(((-1))^n) QPochhammer[2,2,n]-QFactorial[n,2],n∈Integers]

bug(336808)

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

Table[QFactorial[n,2],{n,20}]

QFactorial[Range[10],2]

DominationNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"DominationNumber"],{n,20}]

Demaio and Tran (2013)

DominationPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"DominationPolynomial"][x],{n,k0=1,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,20}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"EdgeChromaticNumber"],{n,20}]

EdgeConnectivity

Table[GraphData[{"TriangularHoneycombRook",n},"EdgeConnectivity"],{n,20}]

EdgeCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"EdgeCount"],{n,20}]

FindSequenceFunction[seq,n]//Factor

Binomial[n+1,3]

EdgeCoverCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"EdgeCoverCount"],{n,20}]

EdgeCoverNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"EdgeCoverNumber"],{n,20}]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"EdgeCoverPolynomial"][x],{n,k0=1,kmax=10}])

EulerianCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"EulerianCycleCount"],{n,20}]

FaceCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"FaceCount"],{n,20}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"FlowPolynomial"][x],{n,k0=1,kmax=10}])

FractionalChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"FractionalChromaticNumber"],{n,20}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"FractionalEdgeChromaticNumber"],{n,20}]

Girth

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Girth"],{n,20}]

Gonality

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Gonality"],{n,20}]

HamiltonConnected

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonConnected"],{n,20}]

HamiltonDecompositionCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonDecompositionCount"],{n,20}]

Hamiltonian

Table[GraphData[{"TriangularHoneycombRook",n},"Hamiltonian"],{n,20}]

HamiltonianCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonianCycleCount"],{n,20}]

HamiltonianNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonianNumber"],{n,20}]

HamiltonianPathCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonianPathCount"],{n,20}]

HamiltonianWalkCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HamiltonianWalkCount"],{n,20}]

HararyIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HararyIndex"],{n,20}]

FindSequenceFunction[seq,n]//Factor

Binomial[n+1,3]

HexagonCount

Entity["Graph",{"Complete",k}]["HexagonCount"]

60Sum[Binomial[k,6],{k,n}]

seq=Table[GraphData[{"TriangularHoneycombRook",n},"HexagonCount"],{n,20}]

Table[60Binomial[n+1,7],{n,20}]

IdiosyncraticPolynomial

Entity["Graph",{"Complete",k}]["IdiosyncraticPolynomial"][x,y]

Product[(((1+x))^(-1+k)) (1-k+x),{k,n}]

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"IdiosyncraticPolynomial"][x,y],{n,k0=1,kmax=10}])//Column

Entity["Graph",{"Complete",k}]["IdiosyncraticPolynomial"][x,y]

Sum[(((1+x))^(-1+k)) (1-k+x),{k,n}]//FullSimplify

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

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IndependenceNumber"],{n,20}]

Demaio and Tran (2013)

IndependencePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"IndependencePolynomial"][x],{n,k0=1,kmax=10}])//Column

Product[1+k x,{k,n}]//FullSimplify

Table[(x^n) Pochhammer[1+(1/x),n],{n,20}]//Factor//Column

IndependenceRatio

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IndependenceRatio"],{n,20}]

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A289897

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IndependentEdgeSetCount"],{n,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

Table[Product[(I/Sqrt[2])^k HermiteH[k,-I/Sqrt[2]],{k,n}],{n,20}]

Product[(I/Sqrt[2])^k HermiteH[k,-I/Sqrt[2]],{k,n}]

Product[(I/Sqrt[2])^k,{k,n}]//FullSimplify

Table[(((-(1/2)))^((1/4) n (1+n)))Product[ HermiteH[k,-I/Sqrt[2]],{k,n}],{n,20}]

Binomial[n+1,2]/2

seq=Table[(((-(1/2)))^(Binomial[n+1,2]/2))Product[HermiteH[k,-I/Sqrt[2]],{k,n}],{n,30}]

hyper2F0([-n/2,(1-n)/2],[],2)

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

Table[Product[HypergeometricPFQ[{-k/2,(1-k)/2},{},2],{k,n}],{n,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

IndependentVertexSetCount

A000142

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IndependentVertexSetCount"],{n,20}]

FindSequenceFunction[seq,n]//FunctionExpand//FullSimplify

Table[(n+1)!,{n,20}]

IntersectionNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"IntersectionNumber"],{n,20}]

KirchhoffIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"KirchhoffIndex"],{n,20}]

KirchhoffSumIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"KirchhoffSumIndex"],{n,20}]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"TriangularHoneycombRook",n},"LaplacianMatrix"],{n,5}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"LaplacianPolynomial"][x],{n,k0=1,kmax=10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

(x^n)Product[(((x-k))^(k-1)),{k,n}]

(poly2=Table[(x^n)Product[(((x-k))^(k-1)),{k,n}],{n,20}])//Column

poly/poly2

LeafCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"LeafCount"],{n,20}]

LongestCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"LongestCycleCount"],{n,20}]

FindSequenceFunction[seq,n]

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

Table[Piecewise[{{0,n==1||n==2}},Gamma[n]/2],{n,20}]

LovaszNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"LovaszNumber"],{n,20}]

MatchingGeneratingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"MatchingGeneratingPolynomial"][x],{n,k0=1,kmax=10}])//Column

Entity["Graph",{"Complete",k}]["MatchingGeneratingPolynomial"][x]

Product[(2^(k/2)) (((-1/x))^(-k/2)) HypergeometricU[-(k/2),(1/2),-(1/(2 x))],{k,n}]

Table[Product[(2^(k/2)) (((-1/x))^(-k/2)) HypergeometricU[-(k/2),(1/2),-(1/(2 x))],{k,n}],{n,20}]//Factor//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,20}]

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MatchingNumber"],{n,20}]

FindSequenceFunction[seq,n]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"MatchingPolynomial"][x],{n,k0=1,kmax=10}])//Column

Entity["Graph",{"Complete",k}]["MatchingPolynomial"][x]

Product[(2^(-k/2)) HermiteH[k,(x/(Sqrt[2]))],{k,n}]

Table[Product[(2^(-k/2)) HermiteH[k,(x/(Sqrt[2]))],{k,n}],{n,20}]//Factor//Column

MaximalCliqueCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximalCliqueCount"],{n,20}]

MaximalIndependentEdgeSetCount

A289900

== MaximumIndependentEdgeSetCount for n > 1

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximalIndependentEdgeSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]

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

FindGeneratingFunction[DeleteMissing@Drop[seq,2],x]

Table[Product[(2Ceiling[k/2]-1)!!,{k,n}],{n,20}]

Table[(2Ceiling[k/2]-1),{k,2,10,2}]

Table[(2Ceiling[k/2]-1),{k,1,10,2}]

Table[Product[If[Mod[k,2]==0,k-1,k]!!,{k,n}],{n,20}]

Table[Product[(k-Mod[k-1,2])!!,{k,n}],{n,20}]

Product[If[Mod[k,2]==0,k-1,k]!!,{k,n}]

Product[(k-1)!!,{k,2,n,2}]

Product[k!!,{k,1,n,2}]

Table[(1/(E^(1/8)))(2^(-(1/24)+(1/2) Floor[(n/2)] (1+Floor[(n/2)]))) (Glaisher^(3/2)) (π^(-(1/4)-(1/2) Floor[(n/2)])) BarnesG[(3/2)+Floor[(n/2)]](((2^(-(1/24)+(1/2) (1+Floor[(1/2) (-1+n)]) (2+Floor[(1/2) (-1+n)]))) (Glaisher^(3/2)) (π^((1/4) (-3-2 Floor[(1/2) (-1+n)]))) BarnesG[(5/2)+Floor[(1/2) (-1+n)]])/(E^(1/8))),{n,20}]//N

FullSimplify[(1/(E^(1/8)))(2^(-(1/24)+(1/2) Floor[(n/2)] (1+Floor[(n/2)]))) (Glaisher^(3/2)) (π^(-(1/4)-(1/2) Floor[(n/2)])) BarnesG[(3/2)+Floor[(n/2)]](((2^(-(1/24)+(1/2) (1+Floor[(1/2) (-1+n)]) (2+Floor[(1/2) (-1+n)]))) (Glaisher^(3/2)) (π^((1/4) (-3-2 Floor[(1/2) (-1+n)]))) BarnesG[(5/2)+Floor[(1/2) (-1+n)]])/(E^(1/8))),Mod[n,2]==#]&/@{0,1}

FullSimplify[((((2^(-(1/12)+(1/4) n (2+n)))  (π^(-(1/2)-(n/2))) ((BarnesG[((3+n)/2)])^2))/(E^(1/4)))/(((2^(1/6)) (E^((1/4) (-1+n (n Log[2]+Log[(4/(π^2))]))))  BarnesG[1+(n/2)] BarnesG[2+(n/2)])/(Sqrt[π])))]

Table[(2^(n (n+2)/4-1/12))  (E^(-1/4))(π^(-(n+1)/2))(Glaisher^3)If[Mod[n,2]==0,((BarnesG[((3+n)/2)])^2),(2^(1/4)) BarnesG[(n/2)+1] BarnesG[(n/2)+2]],{n,20}]

((π^(-(1/2)-(n/2)))^2)

seq=Table[Product[(k-Mod[k-1,2])!!,{k,n}],{n,30}]

FindLinearRecurrence[Rest@seq]//Head

gf=x FindGeneratingFunction[seq,x]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximalIndependentVertexSetCount"],{n,20}]

FindSequenceFunction[seq,n]

MaximumCliqueCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximumCliqueCount"],{n,20}]

MaximumIndependentEdgeSetCount

==MaximalIndependentEdgeSetCount for n > 1

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximumIndependentEdgeSetCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@Drop[seq,2]]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximumIndependentVertexSetCount"],{n,20}]

FindSequenceFunction[seq,n]

MaximumLeafNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MaximumLeafNumber"],{n,20}]

MeanDistance

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MeanDistance"],{n,20}]

MinimalDominatingSetCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimalDominatingSetSignature

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimalDominatingSetSignature"],{n,20}]//Column

FindSequenceFunction[DeleteMissing@seq,n]

MinimumCliqueCoveringCount

Probably 1

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumCliqueCoveringCount"],{n,20}]

MinimumDistinguishingLabelingCount

A002109

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumDistinguishingLabelingCount"],{n,20}]

Table[Hyperfactorial[n],{n,20}]

MinimumDominatingSetCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumDominatingSetCount"],{n,20}]

FindSequenceFunction[seq,x]

MinimumEdgeCoverCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumEdgeCoverCount"],{n,20}]

MolecularTopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MolecularTopologicalIndex"],{n,20}]

Nonhamiltonian

Table[GraphData[{"TriangularHoneycombRook",n},"Nonhamiltonian"],{n,20}]

NormalizedLaplacianMatrix

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

OddChordlessCycleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"OddChordlessCycleCount"],{n,20}]

PathCount

A002807

Entity["Graph",{"Complete",k}]["PathCount"]

Sum[(1/2) E (-1+k) k Gamma[-1+k,1],{k,n}]

Table[Sum[(1/2) E (-1+k) k Gamma[-1+k,1],{k,n}],{n,20}]

Sum[(k-1)!Binomial[n,k]/2,{k,3,n}]

n/4*(2*HypergeometricPFQ[{1,1,1-n},{2},-1]-n-1)/.n->n+1

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

seq=Table[GraphData[{"TriangularHoneycombRook",n},"PathCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

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

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

PathPolynomial

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

Table[Sum[(x^(k-1))(k-1)!Binomial[n+1,k]/2,{k,3,n+1}],{n,20}]//Factor//Column

Sum[(x^(k-1))(k-1)!Binomial[n+1,k]/2,{k,3,n+1}]

Table[(1/4) (1+n) (-2-n+2 HypergeometricPFQ[{1,1,-n},{2},-x]),{n,20}]//Factor//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

PentagonCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"PentagonCount"],{n,20}]

Entity["Graph",{"Complete",k}]["PentagonCount"]

Sum[(1/10) (-4+k) (-3+k) (-2+k) (-1+k) k,{k,n}]

12Binomial[n+1,6]//FunctionExpand

Perfect

Table[GraphData[{"TriangularHoneycombRook",n},"Perfect"],{n,20}]

ProjectivePlaneCrossingNumber

Probably 0

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ProjectivePlaneCrossingNumber"],{n,20}]

Radius

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Radius"],{n,20}]

RankPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"RankPolynomial"][x,y],{n,k0=1,kmax=5}])//Column

Entity["Graph",{"Complete",k}]["RankPolynomial"]

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,20}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

Entity["Graph",{"Complete",k}]["RectilinearCrossingNumber"]

seq=Table[GraphData[{"TriangularHoneycombRook",n},"RectilinearCrossingNumber"],{n,20}]

Table[Sum[Entity["Graph",{"Complete",k}]["RectilinearCrossingNumber"],{k,n}],{n,20}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"ReliabilityPolynomial"][x],{n,k0=1,kmax=5}])//Column

Entity["Graph",{"Complete",k}]["ReliabilityPolynomial"]

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,20}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"SigmaPolynomial"][x],{n,k0=1,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,20}]

FullSimplify[poly2-poly]

SpanningTreeCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"SpanningTreeCount"],{n,20}]

SquareCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"SquareCount"],{n,20}]

Entity["Graph",{"Complete",k}]["SquareCount"]

Sum[(1/8) (-3+k) (-2+k) (-1+k) k,{k,n}]

3Binomial[n+1,5]//FunctionExpand

StabilityIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"StabilityIndex"],{n,20}]

FindSequenceFunction[seq,n]

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

TopologicalIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"TopologicalIndex"],{n,20}]

ToroidalCrossingNumber

Probably 0

seq=Table[GraphData[{"TriangularHoneycombRook",n},"ToroidalCrossingNumber"],{n,20}]

Traceable

Table[GraphData[{"TriangularHoneycombRook",n},"Traceable"],{n,20}]

TriangleCount

seq=Table[GraphData[{"TriangularHoneycombRook",n},"TriangleCount"],{n,20}]

FindSequenceFunction[seq,n]//Factor

Binomial[n+1,4]

TuttePolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"TuttePolynomial"][x,y],{n,k0=1,kmax=10}])//Column

Entity["Graph",{"Complete",k}]["TuttePolynomial"]

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,20}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"TriangularHoneycombRook",n},"Untraceable"],{n,20}]

VertexConnectivity

Table[GraphData[{"TriangularHoneycombRook",n},"VertexConnectivity"],{n,20}]

VertexCount

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

FindSequenceFunction[seq,n]

Binomial[n+1,2]

VertexCoverNumber

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"TriangularHoneycombRook",n},"VertexCoverNumber"],{n,20}]

FindSequenceFunction[seq,n]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"TriangularHoneycombRook",n},"VertexCoverPolynomial"][x],{n,k0=1,kmax=10}])//Column

Table[(x^(Binomial[n,2]-1))(((-1))^(n+1))FactorialPower[-x,n+1]//FunctionExpand//Factor,{n,20}]//Column

VertexDegrees

seq=Table[GraphData[{"TriangularHoneycombRook",n},"VertexDegrees"],{n,20}]

WienerIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"WienerIndex"],{n,20}]

WienerSumIndex

seq=Table[GraphData[{"TriangularHoneycombRook",n},"WienerSumIndex"],{n,20}]

GraphData

GraphDataClassString

GraphDataClassString["TriangularHoneycombRook"]

GraphDataString

file=OpenWrite["~/tri-honeycomb-rook.m"]

Square

Do[WriteString[file,GraphDataString[
TriangularHoneycombRookGraph[n],
"Name"->ToString[n]<>"-triangular honeycomb rook graph",
"StandardName"->{"TriangularHoneycombRook",n},
"Classes"->{"TriangularHoneycombRook"},
"NotationRules"->{"TriangularHoneycombRook"->n},
"Skip"->{"ABCEnergy","ABCSpectralRadius","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","BetweennessCentralities","BurningNumber","CharacteristicPolynomial","Circumference","CliqueCoveringNumber","CoboundaryPolynomial","ConnectedDominatingSets","ConstructionNumber","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","Curvatures","CyclePolynomial","CyclicEdgeConnectivity","DetourMatrix","DominatingSets","EigenvectorCentralities","Embeddings3D","Energy","Genus","GracefulLabeling","HadwigerNumber","HamiltonianPaths","HamiltonianWalk","HamiltonianWalks","HITSCentralities","IdiosyncraticPolynomial","IndependencePolynomial","IrredundacePolynomial","IrredundantSets","KatzCentralities","LinklesslyEmbeddable","LowerMatchingNumber","MatchingGeneratingPolynomial","MatchingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSet","MaximalIndependentVertexSets","MaximumIndependentEdgeSetCount","MaximumIndependentEdgeSets","MaximumIndependentVertexSet","MaximumIndependentVertexSetCount","MaximumIndependentVertexSets","MaximumIrredundantSets","MetricDimension","Meyniel","MinimumCoveringsByMaximalCliques","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","MinimumEdgeCuts","MinimumPathCovering","MinimumVertexCover","MinimumVertexCut","MinimumTotalDominatingSet","MinimumWeightFractionalEdgeColoring","MinimumWeightFractionalColoring","Minors","Outerplanar","PageRankCentralities","PathCoveringNumber","PathPolynomial","PathSignature","Paths","ProjectivePlanar","QuadraticEmbeddingConstant","RandicEnergy","RandicSpectralRadius","RectilinearCrossingCount","Skewness","SomborEnergy","SomborSpectralRadius","SpanningTreeCount","TopologicalIndex","TotalDominationPolynomial","TuttePolynomial","VertexCuts"},
TimeConstraint->10
]],{n,11,15}]