Alkane Graph

Author

Eric W. Weisstein
July 19, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Graphics

n-alkane graphs

GraphicsGrid[Partition[Table[Show[GraphData[{"Alkane",n}]//StyleGraphs[#,VertexSize->{"Scaled",.05}]&,PlotLabel->Style[If[n==1,"methane graph",GraphData[{"Alkane",n},"Name"]],Directive[14,Black,Italic,FontFamily->Times]]],{n,5}],5],Alignment->Top,ImageSize->600,Dividers->All]

Alkane isomer graphs

GraphicsRow[{
Show[GraphData["ButaneGraph"]//StyleGraphs[#,VertexSize->{"Scaled",.03}]&,
PlotLabel->Style["n\[Hyphen]butane graph",Directive[14,Black,Italic,FontFamily->Times]]],
Show[GraphData["IsobutaneGraph"]//StyleGraphs[#,VertexSize->{"Scaled",.03}]&,
PlotLabel->Style["isobutane graph",Directive[14,Black,Italic,FontFamily->Times]]]},Dividers->All,ImageSize->400]

Embeddings

IntegerCoordinates

GraphData[#,"Graph","IntegerCoordinates"][[1]]&/@anes

UnitDistance

GraphData[#,"UnitDistance"]&/@anes

GraphData[#,"Graph","UnitDistance"][[1]]&/@anes

UnitDistance/Grid

Length/@(GraphData[#,"Embeddings"]&/@anes)

GraphData[#,"V"]&/@anes

AlkaneGraph[2,VertexLabels->"Name"]

RecognizeGraph/@Table[AlkaneGraph[n],{n,10}]

Trees

Graph[GraphData[#,"Edges"]]&/@anes

GraphData[#,"Graph","All"][[-1]]&/@anes

Properties

Acyclic

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

AdjacencyMatrixCount

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

FindSequenceFunction[seq,n]

Table[(((3 n+2)!)/(24((2^(n-1))+Floor[(2^(n-2))]))),{n,10}]

Arboricity

Table[GraphData[{"Alkane",n},"Arboricity"],{n,10}]

ArcTransitive

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

ArcTransitivity

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

ArticulationVertexCount

l=Table[GraphData[{"Alkane",n},"ArticulationVertexCount"],{n,10}]

AutomorphismCount

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

FindSequenceFunction[seq,n]//FullSimplify

seq/24

24Table[(2^(n-1))+Floor[(2^(n-2))],{n,10}]

BalabanIndex

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

FindSequenceFunction[seq,n]//Factor

Bandwidth

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

BlockCount

seq=Table[GraphData[{"Alkane",n},"BlockCount"],{n,10}]

FindSequenceFunction[seq,n]

BridgeCount

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

FindSequenceFunction[seq,n]

BurningNumber

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

CharacteristicPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

<<MathWorld`Polynomial`

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]

Chordal

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

ChordCount

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

Chordless

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

ChordlessCycleCount

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

Chords

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

ChromaticallyUnique

seq=Table[GraphData[{"Alkane",n},"ChromaticallyUnique"],{n,10}]

ChromaticInvariant

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

ChromaticNumber

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

ChromaticPolynomial

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

Circumference

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

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]//Factor

CliqueCoveringNumber

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

FindSequenceFunction[seq,n]

CliqueNumber

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

CliquePolynomial

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

CochromaticGraphNames

Table[GraphData[{"Alkane",n},"CochromaticGraphNames"],{n,10}]

Select[GraphData["Alkane"],GraphData[#,"ChromaticallyUnique"]=!=False&]

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

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

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedDominationNumber

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

ConnectedDominationPolynomial

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

ConnectedInducedSubgraphCount

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

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedInducedSubgraphPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

CrossingNumber

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

CycleCount

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

CyclePolynomial

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

Cyclic

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

CyclomaticNumber

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

Degeneracy

seq=Table[GraphData[{"Alkane",n},"Degeneracy"],{n,10}]

DetourIndex

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

FindSequenceFunction[seq,n]//Factor

DetourPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Diameter

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

Dimension

seq=Table[GraphData[{"Alkane",n},"Dimension"],{n,10}]

DistancePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

DistinguishingNumber

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

DomaticNumber

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

DominatingSetCount

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

SeriesCoefficient[((x (-17+4 x))/(-1+5 x)),{x,0,n}]

Table[81 (5^(n-2)),{n,10}]

Table[Piecewise[{{17,n==1}},81 (5^(n-2))],{n,10}]

DominationNumber

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

DominationPolynomial

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

EdgeChromaticNumber

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

EdgeConnectivity

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

EdgeCount

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

FindSequenceFunction[seq,n]

EdgeCoverCount

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

FindSequenceFunction[seq,n]

EdgeCoverNumber

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

FindSequenceFunction[seq,n]

EdgeCoverPolynomial

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

EulerianCycleCount

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

FaceCount

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

FlowPolynomial

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

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

Girth

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

GlobalClusteringCoefficient

seq=Table[GraphData[{"Alkane",n},"GlobalClusteringCoefficient"],{n,10}]

Gonality

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

GracefulLabelingCount

seq=Table[GraphData[{"Alkane",n},"GracefulLabelingCount"],{n,10}]

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

FindSequenceFunction[seq,n]

HamiltonianPathCount

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

HamiltonianWalkCount

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

HararyIndex

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

FindSequenceFunction[seq,n]//FullSimplify

HexagonCount

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

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"Alkane",n},"IdiosyncraticPolynomial"],{n,k0=1,kmax=10}]/.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]

IndependenceNumber

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

FindSequenceFunction[seq,n]

IndependencePolynomial

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

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

IndependenceRatio

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

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A180148

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

FindLinearRecurrence[DeleteMissing[seq]]

RSolve[{3a[n-1]+a[n-2]==a[n],a[0]==2,a[1]==5},a[n],n]

FullSimplify[%%]

LinearRecurrence[{3,1},{5,7},20]

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

Table[(3-(Sqrt[13]))^n//Expand//Factor,{n,10}]

(1/13) (2^(-n)) ((13-2 (Sqrt[13])) (((3-(Sqrt[13])))^(n+1))+(((3+(Sqrt[13])))^n) (13+2 (Sqrt[13])))

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

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

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

IndependentVertexSetCount

A228602

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

FindLinearRecurrence[DeleteMissing[seq]]

LinearRecurrence[{5,-8,4},{3,11,33},20]

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

Table[(1/8) ((12-11 (Sqrt[2])) (((2-2 (Sqrt[2])))^n)+(((2 (1+(Sqrt[2]))))^n) (12+11 (Sqrt[2]))),{n,20}]//Expand

LinearRecurrence[{4,4},{17,80},20]

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

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((17+12 x)/(1-4 x-4 (x^2))),{x,0,20}],x]

IntersectionNumber

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

IrredundanceNumber

seq=Table[GraphData[{"Alkane",n},"IrredundanceNumber"],{n,10}]

IrredundancePolynomial

(poly=DeleteMissing@Table[GraphData[{"Alkane",n},"IrredundancePolynomial"],{n,10}]/.f_Function:>f[x])//Column

CoefficientList[poly,x]//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindLinearRecurrence[Rest@poly]

IrredundantSetCount

seq=Table[GraphData[{"Alkane",n},"IrredundantSetCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

coef=FindLinearRecurrence[DeleteMissing@Drop[seq,3]]

gf=x FindGeneratingFunction[seq,x]

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

IsolatedPointCount

l=Table[GraphData[{"Alkane",n},"IsolatedPointCount"],{n,10}]

KirchhoffIndex

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

FindSequenceFunction[seq,n]//Factor

KirchhoffSumIndex

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

FindSequenceFunction[seq,n]//Factor

LaplacianMatrix

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

LaplacianPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

LeafCount

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

FindSequenceFunction[seq,n]

LongestCycleCount

Same as HC

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

LongestPathCount

seq=Table[GraphData[{"Alkane",n},"LongestPathCount"],{n,10}]

LongestPathLength

seq=Table[GraphData[{"Alkane",n},"LongestPathLength"],{n,10}]

LovaszNumber

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

LowerIndependenceNumber

seq=Table[GraphData[{"Alkane",n},"LowerIndependenceNumber"],{n,10}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

LowerMatchingNumber

seq=Table[GraphData[{"Alkane",n},"LowerMatchingNumber"],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

MatchingGeneratingPolynomial

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

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

MatchingNumber

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

FindSequenceFunction[seq,n]

MatchingPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

MaximalCliqueCount

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

FindSequenceFunction[seq,n]

MaximalIndependentEdgeSetCount

A052542

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

2Table[Fibonacci[n,2],{n,0,20}]

A000129

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

seq/2

FindSequenceFunction[seq,n]//FullSimplify

FindLinearRecurrence[seq]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximalIndependentVertexSetCount

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

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

MaximalIrredundantSetCount

seq=Table[GraphData[{"Alkane",n},"MaximalIrredundantSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

MaximumCliqueCount

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

FindSequenceFunction[seq,n]

MaximumIndependentEdgeSetCount

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

gf=x FindGeneratingFunction[seq,x]

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

MaximumIndependentVertexSetCount

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

MaximumIrredundantSetCount

seq=Table[GraphData[{"Alkane",n},"MaximumIrredundantSetCount"],{n,10}]

MaximumLeafNumber

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

FindSequenceFunction[seq,n]

MaximumVertexDegree

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

MeanDistance

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

FindSequenceFunction[seq,n]//Factor

MeanClusteringCoefficient

seq=Table[GraphData[{"Alkane",n},"MeanClusteringCoefficient"],{n,10}]

MetricDimension

seq=Table[GraphData[{"Alkane",n},"MetricDimension"],{n,10}]

MinimalDominatingSetCount

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

MinimalEdgeCoverCount

seq=Table[GraphData[{"Alkane",n},"MinimalEdgeCoverCount"],{n,10}]

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"Alkane",n},"MinimalTotalDominatingSetCount"],{n,10}]

MinimalTotalDominationPolynomial

poly=Table[GraphData[{"Alkane",n},"MinimalTotalDominationPolynomial"]/.f_Function:>f[x],{n,10}]

MinimumConnectedDominatingSetCount

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

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"Alkane",n},"MinimumCoveringsByMaximalCliquesCount"],{n,10}]

MinimumDominatingSetCount

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

MinimumEdgeCoverCount

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

MinimumLeafNumber

seq=Table[GraphData[{"Alkane",n},"MinimumLeafNumber"],{n,10}]

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"Alkane",n},"MinimumTotalDominatingSetCount"],{n,10}]

MinimumVertexCutCount

seq=Table[GraphData[{"Alkane",n},"MinimumVertexCutCount"],{n,10}]

MinimumVertexDegree

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

MolecularTopologicalIndex

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]//Factor

gf=x FindGeneratingFunction[seq,x]//Factor

Nonhamiltonian

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

OddChordlessCycleCount

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

PathCoveringNumber

Computed up to n=3; inferred for rest

Table[GraphData[{"Alkane",n},"PathCoveringNumber"],{n,10}]

PathCount

A060544

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

(1/2) (2+9 n+9 (n^2))/.n->n-1//Factor

Table[(3n-1)(3n-2)/2,{n,20}]

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

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

PathCoveringNumber

seq=Table[GraphData[{"Alkane",n},"PathCoveringNumber"],{n,10}]

PathPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

PebblingNumber

seq=Table[GraphData[{"Alkane",n},"PebblingNumber"],{n,10}]

PentagonCount

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

Perfect

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

ProjectivePlaneCrossingNumber

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

RadioLabelingCount

seq=Table[GraphData[{"Alkane",n},"RadioLabelingCount"],{n,10}]

RadioNumber

seq=Table[GraphData[{"Alkane",n},"RadioNumber"],{n,10}]

Radius

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

RankPolynomial

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

RectilinearCrossingNumber

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

ReliabilityPolynomial

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

ScrambleNumber

seq=Table[GraphData[{"Alkane",n},"ScrambleNumber"],{n,10}]

SigmaPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SpanningTreeCount

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

SpectralRadius

seq=Table[GraphData[{"Alkane",n},"SpectralRadius"],{n,10}]

SquareCount

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

StabilityIndex

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

FindSequenceFunction[seq[[2;;;;2]],n]

gf=x FindGeneratingFunction[seq,x]

SymmetricallyDistinctVertexPairCount

seq=Table[GraphData[{"Alkane",n},"SymmetricallyDistinctVertexPairCount"],{n,10}]

FindSequenceFunction[seq,n]//FullSimplify

TopologicalIndex

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

FindSequenceFunction[seq,n]//FullSimplify

ToroidalCrossingNumber

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

TotalDominatingSetCount

seq=Table[GraphData[{"Alkane",n},"TotalDominatingSetCount"],{n,10}]

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

Table[(4^(n+1)),{n,2,10}]

TotalDominationNumber

seq=Table[GraphData[{"Alkane",n},"TotalDominationNumber"],{n,10}]

Traceable

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

Triameter

seq=Table[GraphData[{"Alkane",n},"Triameter"],{n,10}]

TriangleCount

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

TuttePolynomial

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

Untraceable

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

UpperDominationNumber

Table[GraphData[{"Alkane",n},"UpperDominationNumber"],{n,10}]

UpperIrredundanceNumber

Table[GraphData[{"Alkane",n},"UpperIrredundanceNumber"],{n,10}]

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

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

VertexCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

VertexDegrees

Table[Tally@Sort@GraphData[{"Alkane",n},"VertexDegrees"],{n,10}]//Column

VertexTransitive

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

WienerIndex

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

FindSequenceFunction[seq,n]//Factor

WienerSumIndex

A060544

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

FindSequenceFunction[seq,n]