Helm

Author

Eric W. Weisstein
June 19, 2023

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

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

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

Sources

Seoud and Youssef

Seoud and Youssef.  "Harmonious labelling of helms and related graphs".  Unpublished manuscript.  January 2017.

https://www.researchgate.net/publication/312218305_Harmonious_labelling_of_helms_and_related_graphs

Graphic

GraphicsRow[Show[HelmGraph[#,VertexSize->{"Scaled",.07},System`VertexStyle->Red,System`EdgeStyle->Black],PlotLabel->Style[Subscript[H,#],FontFamily->"Times"]]&/@Range[3,8],ImageSize->500]

Properties

Acyclic

Table[GraphData[{"Helm",n},"Acyclic"],{n,3,20}]

AdjacencyMatrixCount

A174549

seq=Table[GraphData[{"Helm",n},"AdjacencyMatrixCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[Take[seq,10],n-2]//FullSimplify

Table[(2n+1)Gamma[2n],{n,3,20}]

Arboricity

Table[GraphData[{"Helm",n},"Arboricity"],{n,3,20}]

ArcTransitive

Table[GraphData[{"Helm",n},"ArcTransitive"],{n,3,20}]

ArcTransitivity

Table[GraphData[{"Helm",n},"ArcTransitivity"],{n,3,20}]

AutomorphismCount

seq=Table[GraphData[{"Helm",n},"AutomorphismCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

BalabanIndex

seq=Table[GraphData[{"Helm",n},"BalabanIndex"],{n,3,20}]

Bandwidth

Table[GraphData[{"Helm",n},"Bandwidth"],{n,3,20}]

BridgeCount

seq=Table[GraphData[{"Helm",n},"BridgeCount"],{n,3,20}]

BurningNumber

seq=Table[GraphData[{"Helm",n},"BurningNumber"],{n,3,20}]

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"Helm",n},"CharacteristicPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

p=.

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

ChordlessCycleCount

seq=Table[GraphData[{"Helm",n},"ChordlessCycleCount"],{n,3,20}]

FindSequenceFunction[seq,n]

ChromaticInvariant

seq=Table[GraphData[{"Helm",n},"ChromaticInvariant"],{n,3,20}]

ChromaticNumber

seq=Table[GraphData[{"Helm",n},"ChromaticNumber"],{n,3,20}]

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"Helm",n},"CharacteristicPolynomial"],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"Helm",n},"Circumference"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

Classes

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

CliqueCount

seq=Table[GraphData[{"Helm",n},"CliqueCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

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

CliqueCoveringNumber

seq=Table[GraphData[{"Helm",n},"CliqueCoveringNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

CliqueNumber

seq=Table[GraphData[{"Helm",n},"CliqueNumber"],{n,3,20}]

CliquePolynomial

(poly=Factor[Table[GraphData[{"Helm",n},"CliquePolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

poly2=Table[(1+x)Piecewise[{{ (1+6 x+3 (x^2)+(x^3)),n==3}},(1+2n x+n x^2)],{n,k0,kmax}]

FullSimplify[poly2-poly]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Helm",n},"ComplementOddChordlessCycleCount"],{n,3,20}]

ConnectedComponentCount

seq=Table[GraphData[{"Helm",n},"ConnectedComponentCount"],{n,3,20}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"Helm",n},"ConnectedDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing[seq],n-2]

ConnectedDominationNumber

seq=Table[GraphData[{"Helm",n},"ConnectedDominationNumber"],{n,3,20}]

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedDominationPolynomial

(poly=DeleteCases[Factor@Table[GraphData[{"Helm",n},"ConnectedDominationPolynomial"][x],{n,k0=3,20}],_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>2]

ConnectedInducedSubgraphCount

A286184

seq=Table[GraphData[{"Helm",n},"ConnectedInducedSubgraphCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing[seq]]

FullSimplify[FindSequenceFunction[DeleteMissing[seq],n-2],n∈Integers]

LinearRecurrence[{9,-31,51,-40,12},{6,19,56,157,430},20]

(((x+(x^2)))^n)+x (((1+x+(x^2)))^n)+((n (-2 x+(x^3)+(((x+(x^2)))^n)))/(-1+x+(x^2)))/.x->1

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"Helm",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=3,kmax=20}]/.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[DeleteMissing@poly,z]]

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

poly2=Table[(((x+(x^2)))^n)+x (((1+x+(x^2)))^n)+((n (-2 x+(x^3)+(((x+(x^2)))^n)))/(-1+x+(x^2))),{n,k0,kmax}]

FullSimplify[Take[poly2-poly,11]]

CrossingNumber

seq=Table[GraphData[{"Helm",n},"CrossingNumber"],{n,3,20}]

CycleCount

seq=Table[GraphData[{"Helm",n},"CycleCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Limit[(x^n)+((n (x^2) (-x+(x^n)))/(-1+x)),x->1]

CyclePolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"CyclePolynomial"],{n,k0=3,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>2]

Cyclic

Table[GraphData[{"Helm",n},"Cyclic"],{n,3,20}]

CyclomaticNumber

seq=Table[GraphData[{"Helm",n},"CyclomaticNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]

DetourIndex

seq=Table[GraphData[{"Helm",n},"DetourIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

DetourMatrix

MatrixForm/@Table[GraphData[{"Helm",n},"DetourMatrix"],{n,3,6}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,3,20}]

DetourPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"DetourPolynomial"][x],{n,k0=3,kmax=20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

FindSequenceFunction[{90,208,400,684,1078},n-2]//Simplify

FindSequenceFunction[{96,204,370,606,924},n-2]//Factor

FindSequenceFunction[{16,30,48,70,96},n-2]//Factor

poly2=Table[-(((2n+2(n+1)x+(x^2)))^(n-1))(-(n^2) (1+3 n)-x n (-1+5 n+2 (n^2))-2 (-1+n) (1+n)(x^2)+(x^3)),{n,k0,kmax}]

FullSimplify[poly2-poly]

Diameter

seq=Table[GraphData[{"Helm",n},"Diameter"],{n,3,20}]

DistanceMatrix

MatrixForm/@Table[GraphData[{"Helm",n},"DistanceMatrix"],{n,3,10}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,3,20}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"DistancePolynomial"][x],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

DistinguishingNumber

seq=Table[GraphData[{"Helm",n},"DistinguishingNumber"],{n,3,20}]

DomaticNumber

seq=Table[GraphData[{"Helm",n},"DomaticNumber"],{n,3,20}]

DominatingSetCount

A048473

seq=Table[GraphData[{"Helm",n},"DominatingSetCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

Table[2 3^n-1,{n,20}]

2 3^Range[20]-1

DominationNumber

seq=Table[GraphData[{"Helm",n},"DominationNumber"],{n,3,20}]

DominationPolynomial

A213666 with initial 0 added to each row

(poly=Factor@Table[GraphData[{"Helm",n},"DominationPolynomial"][x],{n,k0=3,kmax=20}]/.f_Function:>f[x])//Column

CoefficientList[Take[poly,4],x]

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"Helm",n},"EdgeChromaticNumber"],{n,3,20}]

EdgeConnectivity

Table[GraphData[{"Helm",n},"EdgeConnectivity"],{n,3,20}]

EdgeCount

seq=Table[GraphData[{"Helm",n},"EdgeCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

EdgeCoverCount

seq=Table[GraphData[{"Helm",n},"EdgeCoverCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

EdgeCoverNumber

seq=Table[GraphData[{"Helm",n},"EdgeCoverNumber"],{n,3,20}]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"EdgeCoverPolynomial"][x],{n,k0=3,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>2]

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

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"Helm",n},"EulerianCycleCount"],{n,3,20}]

FaceCount

seq=Table[GraphData[{"Helm",n},"FaceCount"],{n,3,20}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"FlowPolynomial"][x],{n,k0=3,kmax=20}]/.f_Function:>f[x])

FractionalChromaticNumber

seq=Table[GraphData[{"Helm",n},"FractionalChromaticNumber"],{n,3,20}]

FullSimplify[FindSequenceFunction[seq,n-2],n∈Integers]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Helm",n},"FractionalEdgeChromaticNumber"],{n,3,20}]

Girth

seq=Table[GraphData[{"Helm",n},"Girth"],{n,3,20}]

Gonality

seq=Table[GraphData[{"Helm",n},"Gonality"],{n,3,20}]

HamiltonConnected

seq=Table[GraphData[{"Helm",n},"HamiltonConnected"],{n,3,20}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Helm",n},"HamiltonDecompositionCount"],{n,3,20}]

Hamiltonian

Table[GraphData[{"Helm",n},"Hamiltonian"],{n,3,20}]

HamiltonianCycleCount

seq=Table[GraphData[{"Helm",n},"HamiltonianCycleCount"],{n,3,20}]

HamiltonianNumber

seq=Table[GraphData[{"Helm",n},"HamiltonianNumber"],{n,3,20}]

HamiltonianPathCount

seq=Table[GraphData[{"Helm",n},"HamiltonianPathCount"],{n,3,20}]

HamiltonianWalkCount

seq=Table[GraphData[{"Helm",n},"HamiltonianWalkCount"],{n,3,20}]

HararyIndex

seq=Table[GraphData[{"Helm",n},"HararyIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

HexagonCount

seq=Table[GraphData[{"Helm",n},"HexagonCount"],{n,3,20}]

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

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

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"IdiosyncraticPolynomial"][x,y],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"Helm",n},"IndependenceNumber"],{n,3,20}]

IndependencePolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"IndependencePolynomial"][x],{n,k0=3,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>2]

IndependenceRatio

seq=Table[GraphData[{"Helm",n},"IndependenceRatio"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

IndependentEdgeSetCount

A286814

seq=Table[GraphData[{"Helm",n},"IndependentEdgeSetCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing[seq]]

FullSimplify[FindSequenceFunction[DeleteMissing[seq],n-2],n∈Integers]

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

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

Table[(((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n),{n,10}]//Simplify

Table[LucasL[n,2],{n,10}]

Table[((Sqrt[2])/4)(-(((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n)),{n,10}]//Simplify

Table[Fibonacci[n,2],{n,10}]

(1/4) ((((1-(Sqrt[2])))^n) (4-(Sqrt[2]) n)+(((1+(Sqrt[2])))^n) (4+(Sqrt[2]) n))

Table[(1/4) ((((1-(Sqrt[2])))^n) (4-(Sqrt[2]) n)+(((1+(Sqrt[2])))^n) (4+(Sqrt[2]) n)),{n,0,20}]//Expand

seq=Table[LucasL[n,2]+n Fibonacci[n,2],{n,0,30}]

FindGeneratingFunction[seq,x]

StringReplace[ToString[%,InputForm]," "->""]

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

LinearRecurrence[{4,-2,-4,-1},{3,10,29,82},{0,20}]

IndependentVertexSetCount

A287594

seq=Table[GraphData[{"Helm",n},"IndependentVertexSetCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing[seq]]

LinearRecurrence[{4,-2,-4},{4,12,28},{0,30}]

FullSimplify[FindSequenceFunction[DeleteMissing[seq],n]]

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

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

Table[(((1-(Sqrt[3])))^n)+(((1+(Sqrt[3])))^n),{n,0,20}]//Expand

Table[(2^n)+(((1-(Sqrt[3])))^n)+(((1+(Sqrt[3])))^n),{n,0,20}]//Expand

seq=Table[(2^n)+(2^(n/2))LucasL[n,Sqrt[2]],{n,0,20}]//Round

gf=FindGeneratingFunction[seq,x]//Simplify

StringReplace[ToString[gf,InputForm]," "->""]

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

IntersectionNumber

seq=Table[GraphData[{"Helm",n},"IntersectionNumber"],{n,3,20}]

IrredundantSetCount

A290588

seq=Table[GraphData[{"Helm",n},"IrredundantSetCount"],{n,3,20}]

seq={35,113,355,1105,3435,10625,32723,100433,307323,937985,2856547,8682897,26349835,79850241};

FindLinearRecurrence[DeleteMissing@seq]

seq=LinearRecurrence[{6,-10,4,-5,6},{5,13,35,113,355},40]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Table[1+2 (3^n)-RootSum[-2-#-2 ((#)^2)+((#)^3)&,#^n&],{n,20}]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

CoefficientList[Series[((-5+17 x-7 (x^2)-13 (x^3))/(-1+6 x-10 (x^2)+4 (x^3)-5 (x^4)+6 (x^5))),{x,0,20}],x]

KirchhoffIndex

seq=Table[GraphData[{"Helm",n},"KirchhoffIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

KirchhoffSumIndex

seq=Table[GraphData[{"Helm",n},"KirchhoffSumIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Helm",n},"LaplacianMatrix"],{n,3,5}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"LaplacianPolynomial"][x],{n,k0=3,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>2]

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

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"Helm",n},"LeafCount"],{n,3,20}]

LongestCycleCount

seq=Table[GraphData[{"Helm",n},"LongestCycleCount"],{n,3,20}]

LovaszNumber

seq=Table[GraphData[{"Helm",n},"LovaszNumber"],{n,3,20}]

MatchingGeneratingPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"MatchingGeneratingPolynomial"][x],{n,k0=3,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>2]

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

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"Helm",n},"MatchingNumber"],{n,3,20}]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"MatchingPolynomial"][x],{n,k0=3,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>2]

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

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"Helm",n},"MaximalCliqueCount"],{n,3,20}]

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

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

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

MaximalIndependentEdgeSetCount

A061705

seq=Table[GraphData[{"Helm",n},"MaximalIndependentEdgeSetCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n-2],n>2&&n∈Integers]

Table[(2^(-n)) ((((1-(Sqrt[5])))^n) +(((1+(Sqrt[5])))^n)),{n,10}]//Expand

Table[LucasL[n],{n,10}]

Table[(((2^(-n)) )/(Sqrt[5]))(-(((1-(Sqrt[5])))^n) +(((1+(Sqrt[5])))^n)),{n,10}]//Expand

Table[Fibonacci[n],{n,10}]

Table[(1/5) (2^(-n)) ((((1-(Sqrt[5])))^n) (5-(Sqrt[5]) n)+(((1+(Sqrt[5])))^n) (5+(Sqrt[5]) n)),{n,0,20}]//Expand

Table[LucasL[n]+n Fibonacci[n],{n,0,20}]

seq=LinearRecurrence[{2,1,-2,-1},{2,5,10,19},{0,20}]

gf=FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

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

{{10,2},{11,300},{12,4125},{13,18480},{14,38610},{15,44044},{16,29575},{17,12000},{18,2890},{19,380},{20,21}}[[All,2]]//Total

MaximalIndependentVertexSetCount

A000032

seq=Table[GraphData[{"Helm",n},"MaximalIndependentVertexSetCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

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

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

Table[LucasL[n],{n,3,20}]

MaximalIrredundantSetCount

A291097

seq=Table[GraphData[{"Helm",n},"MaximalIrredundantSetCount"],{n,3,20}]

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{6,-13,12,-4},{3,8,20,47},20]

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

seq=Table[(2^n)(n/8+1)-n,{n,2,40}]

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

CoefficientList[Series[((3-10 x+11 (x^2)-5 (x^3))/(((1-3 x+2 (x^2)))^2)),{x,0,20}],x]

MaximumCliqueCount

seq=Table[GraphData[{"Helm",n},"MaximumCliqueCount"],{n,3,20}]

FindSequenceFunction[Rest@seq,n-4]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Helm",n},"MaximumIndependentEdgeSetCount"],{n,3,20}]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Helm",n},"MaximumIndependentVertexSetCount"],{n,3,20}]

MaximumLeafNumber

seq=Table[GraphData[{"Helm",n},"MaximumLeafNumber"],{n,3,20}]

MaximumVertexDegree

seq=Table[GraphData[{"Helm",n},"MaximumVertexDegree"],{n,3,20}]

MeanDistance

seq=Table[GraphData[{"Helm",n},"MeanDistance"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"Helm",n},"MinimalConnectedDominatingSetCount"],{n,3,20}]

MinimalDominatingSetCount

seq=Table[GraphData[{"Helm",n},"MinimalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

MinimalDominatingSetSignature

seq=Table[GraphData[{"Helm",n},"MinimalDominatingSetSignature"],{n,3,20}]//Column

MinimalEdgeCoverCount

seq=Table[GraphData[{"Helm",n},"MinimalEdgeCoverCount"],{n,3,20}]

MinimalEdgeCoverSignature

seq=Table[GraphData[{"Helm",n},"MinimalEdgeCoverSignature"],{n,3,20}]//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Helm",n},"MinimumCliqueCoveringCount"],{n,3,20}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Helm",n},"MinimumConnectedDominatingSetCount"],{n,3,20}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Helm",n},"MinimumDistinguishingLabelingCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]

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

MinimumDominatingSetCount

seq=Table[GraphData[{"Helm",n},"MinimumDominatingSetCount"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Helm",n},"MinimumEdgeCoverCount"],{n,3,20}]

MinimumVertexDegree

seq=Table[GraphData[{"Helm",n},"MinimumVertexDegree"],{n,3,20}]

MolecularTopologicalIndex

seq=Table[GraphData[{"Helm",n},"MolecularTopologicalIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

Nonhamiltonian

Table[GraphData[{"Helm",n},"Nonhamiltonian"],{n,3,20}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"Helm",n},"NormalizedLaplacianMatrix"],{n,3,5}]

OddChordlessCycleCount

seq=Table[GraphData[{"Helm",n},"OddChordlessCycleCount"],{n,3,20}]

PathCount

A292001

seq=Table[GraphData[{"Helm",n},"PathCount"],{n,3,20}]

FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{5,-10,10,-5,1},{93,260,615,1278,2401},{-1,1}]

seq=LinearRecurrence[{5,-10,10,-5,1},{3,26,93,260,615},40]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

Table[n(-21+38 n-12 (n^2)+4 (n^3))/3,{n,20}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((-3-11 x+7 (x^2)-25 (x^3))/(((-1+x))^5)),{x,0,20}],x]

PathPolynomial

(poly=Factor[Table[GraphData[{"Helm",n},"PathPolynomial"],{n,k0=3,kmax=20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

PentagonCount

seq=Table[GraphData[{"Helm",n},"PentagonCount"],{n,3,20}]

Perfect

Table[GraphData[{"Helm",n},"Perfect"],{n,3,20}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Helm",n},"ProjectivePlaneCrossingNumber"],{n,3,20}]

Radius

seq=Table[GraphData[{"Helm",n},"Radius"],{n,3,20}]

RankPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"RankPolynomial"][x,y],{n,k0=3,kmax=20}]/.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]]

Simplify[SeriesCoefficient[gf,{z,0,n}],n>2]

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

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"Helm",n},"RectilinearCrossingNumber"],{n,3,20}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"ReliabilityPolynomial"],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"SigmaPolynomial"][x],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

SpanningTreeCount

A004146

seq=Table[GraphData[{"Helm",n},"SpanningTreeCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[LucasL[2n]-2,{n,3,20}]

SquareCount

seq=Table[GraphData[{"Helm",n},"SquareCount"],{n,3,20}]

StabilityIndex

seq=Table[GraphData[{"Helm",n},"StabilityIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

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

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

Table[(((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n),{n,10}]//Expand

Table[LucasL[n,2],{n,10}]

Table[(1/4) (LucasL[n,2] n-2 n Cos[((n π)/2)]-2 (n+4) Sin[((n π)/2)]),{n,3,20}]

TopologicalIndex

seq=Table[GraphData[{"Helm",n},"TopologicalIndex"],{n,3,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

ToroidalCrossingNumber

seq=Table[GraphData[{"Helm",n},"ToroidalCrossingNumber"],{n,3,20}]

Traceable

Table[GraphData[{"Helm",n},"Traceable"],{n,3,20}]

TriangleCount

seq=Table[GraphData[{"Helm",n},"TriangleCount"],{n,3,20}]

TuttePolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"TuttePolynomial"],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"Helm",n},"Untraceable"],{n,3,20}]

VertexConnectivity

Table[GraphData[{"Helm",n},"VertexConnectivity"],{n,3,20}]

VertexCount

seq=Table[GraphData[{"Helm",n},"VertexCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

VertexCoverNumber

Related to but not the same as IndependenceNumber

seq=Table[GraphData[{"Helm",n},"VertexCoverNumber"],{n,3,20}]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"Helm",n},"VertexCoverPolynomial"][x],{n,k0=3,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,k0,kmax}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"Helm",n},"VertexDegrees"],{n,3,20}]

VertexTransitive

Table[GraphData[{"Helm",n},"VertexTransitive"],{n,3,20}]

WienerIndex

seq=Table[GraphData[{"Helm",n},"WienerIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Simplify

WienerSumIndex

seq=Table[GraphData[{"Helm",n},"WienerSumIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

GraphData

GraphClassString

GraphClassString[Select[GraphData["Helm"],GraphData[#,"V"]>0&]]

GraphDataString

Do[Print[GraphDataString[HelmGraph[n],
"Name"->ToString[n]<>"-helm graph",
"StandardName"->{"Helm",n},
"Classes"->{"Helm"},
"NotationRules"->{"Helm"->n},
"Skip"->{"Perfect"},
TimeConstraint->120
]],{n,3,20}]