Alternating Group Graph

Author

Eric W. Weisstein
July 30, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Figure

gs=SortBy[{("AlternatingGroup"/.GraphData[#,"NotationRules"]),#}&/@GraphData["AlternatingGroup"],First]

nicegs=Select[gs,GraphData[#[[2]],"Embeddings"]=!={}&]

GraphicsGrid[Partition[Show[GraphData[#[[2]]]/.g_System`Graph:>SetProperty[g,{VertexSize->If[#[[1]]==2,.01,{"Scaled",.04}],System`VertexStyle->Red,System`EdgeStyle->Black}],Method->None,BaseStyle->Italic,PlotLabel->TextCell[Style[Column[{Subscript["AG",Style[#[[1]],Plain]],GraphData[#[[2]],"Name"]},Alignment->Center],FontFamily->"Times"]]]&/@nicegs,UpTo[3]],ImageSize->450]

Special Cases

Sort[("AlternatingGroup"/.GraphData[#,"NotationRules"])->#&/@GraphData["AlternatingGroup"]]

TextGrid[DeleteCases[Table[{n,RecognizeGraph[AlternatingGroupGraph[n]]},{n,8}],{_,{}}],Dividers->All]

Properties

Acyclic

Table[GraphData[{"AlternatingGroupGraph",n},"Acyclic"],{n,2,6}]

AdjacencyMatrixCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"AdjacencyMatrixCount"],{n,2,6}]

Table[((((1/2) Gamma[1+n])!)/(Piecewise[{{1,n==2}},(n-2)!n!])),{n,2,6}]

Table[(((n!/2)!)/(Piecewise[{{1,n==2}},(n-2)!n!])),{n,2,6}]

Anarboricity

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Anarboricity"],{n,2,6}]

Arboricity

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Arboricity"],{n,2,6}]

ArcTransitive

Table[GraphData[{"AlternatingGroupGraph",n},"ArcTransitive"],{n,2,6}]

ArcTransitivity

Table[GraphData[{"AlternatingGroupGraph",n},"ArcTransitivity"],{n,2,6}]

AutomorphismCount

A175430

seq=Table[GraphData[{"AlternatingGroupGraph",n},"AutomorphismCount"],{n,2,6}]

Table[(n-2)!n!,{n,2,7}]

(seq=Table[GroupOrder[GraphAutomorphismGroup[AlternatingGroupGraph[n]]],{n,9}])//Timing

Table[Piecewise[{{1,n==2}},(n-2)!n!],{n,2,9}]

seq=Table[(n-1)!(n+1)!,{n,30}];

FindLinearRecurrence[seq]//Head

FindSequenceFunction[seq,n]//FullSimplify

seq=Table[Gamma[n] Gamma[2+n],{n,30}];

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

BalabanIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"BalabanIndex"],{n,2,6}]

FindSequenceFunction[seq,n]

Bandwidth

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Bandwidth"],{n,2,6}]

BridgeCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"BridgeCount"],{n,2,6}]

BurningNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"BurningNumber"],{n,2,6}]

CharacteristicPolynomial

Factor@(poly=Table[GraphData[{"AlternatingGroupGraph",n},"CharacteristicPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

CoefficientList[Take[poly,10],x]

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]

ChordCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ChordCount"],{n,2,6}]

Table[GraphData[{"AlternatingGroupGraph",n},"EdgeCount"],{n,2,6}]

Table[(n-1)n!/2,{n,2,6}]

Chordless

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Chordless"],{n,2,6}]

ChordlessCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ChordlessCycleCount"],{n,2,6}]

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

Chords

Length/@Table[GraphData[{"AlternatingGroupGraph",n},"Chords"],{n,2,6}]

ChromaticInvariant

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ChromaticInvariant"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]

ChromaticNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ChromaticNumber"],{n,2,6}]

ChromaticPolynomial

Factor@(poly=Table[GraphData[{"AlternatingGroupGraph",n},"ChromaticPolynomial"],{n,k0=1,kmax=6}]/.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]

Subtract@@@Table[{GraphData[{"AlternatingGroupGraph",n},"ChromaticPolynomial"][x],x ((((-1))^n)-(((-3+x))^n)+(((-2+x))^n)+((((-1))^n) (-3+x)+(((-3+x))^n)) x)},{n,3,13}]//Expand

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

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

Circumference

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Circumference"],{n,2,6}]

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

Table[n!,{n,3,6}]

Classes

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

{"Bicolorable","Bipartite","AlternatingGroupGraph","Class1","Connected","Perfect","Regular","Simple","Traceable","TriangleFree","VertexTransitive","WeaklyPerfect"}

CliqueCount

A139149

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CliqueCount"],{n,2,6}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

Table[((n+1)!+2)/2,{n,2,6}]

CliqueCoveringNumber

Probably A001710

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CliqueCoveringNumber"],{n,2,6}]

Numerator[Range[6]!/2]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n],n>2]

Table[Ceiling[n/2],{n,2,6}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"CliquePolynomial"][x],{n,k0=1,kmax=6}])//Column

CliqueNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CliqueNumber"],{n,2,6}]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ComplementOddChordlessCycleCount"],{n,2,6}]

ConnectedDominatingSetCount

A317481

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ConnectedDominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

ConnectedDominationNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ConnectedDominationNumber"],{n,2,6}]

ConnectedDominationPolynomial

(poly=Table[GraphData[{"AlternatingGroupGraph",n},"ConnectedDominationPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x])//Factor//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]

Subtract@@@Table[{GraphData[{"AlternatingGroupGraph",n},"ConnectedDominationPolynomial"][x],n (x^(-2+n))+n (x^(-1+n))+(x^n)+2 x (-1+(((1+x))^n))+(x^2) (-1+(((1+x))^n))},{n,3,13}]//Expand

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ConnectedInducedSubgraphCount"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimplify

ConnectedInducedSubgraphPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=1,6}]/.f_Function:>f[x])//Factor//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=(z^k0)FindGeneratingFunction[poly,z]

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

FullSimplify[%-poly]

CrossingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CrossingNumber"],{n,2,6}]

Table[GraphData[{"AlternatingGroupGraph",n},"Planar"],{n,2,6}]

CycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CycleCount"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

CyclePolynomial

Factor@(poly=Table[GraphData[{"AlternatingGroupGraph",n},"CyclePolynomial"],{n,k0=1,kmax=6}]/.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=(z^k0)FindGeneratingFunction[poly,z]

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

Cyclic

Table[GraphData[{"AlternatingGroupGraph",n},"Cyclic"],{n,2,6}]

CyclomaticNumber

A317483

seq=Table[GraphData[{"AlternatingGroupGraph",n},"CyclomaticNumber"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimplify

DetourIndex

A296782

seq=Table[GraphData[{"AlternatingGroupGraph",n},"DetourIndex"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

DetourPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"DetourPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x])//Column

Diameter

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Diameter"],{n,2,6}]

FindSequenceFunction[seq,n]

DistancePolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"DistancePolynomial"],{n,k0=1,kmax=6}]/.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]

DistinguishingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"DistinguishingNumber"],{n,2,6}]

DomaticNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"DomaticNumber"],{n,2,6}]

DominationNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"DominationNumber"],{n,2,6}]

DominationPolynomial

(poly=Factor[Table[GraphData[{"AlternatingGroupGraph",n},"DominationPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]

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}]

Simplify[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>2]

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

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

DominatingSetCount

A317484

seq=Table[GraphData[{"AlternatingGroupGraph",n},"DominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

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

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

Table[3 (2^n)-2+RootSum[-1-#1-((#1)^2)+((#1)^3)&,#^n&],{n,2,6}]

EdgeChromaticNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"EdgeChromaticNumber"],{n,2,6}]

EdgeConnectivity

Table[GraphData[{"AlternatingGroupGraph",n},"EdgeConnectivity"],{n,2,6}]

EdgeCount

A181967

seq=Table[GraphData[{"AlternatingGroupGraph",n},"M"],{n,2,6}]

(seq=Table[EdgeCount[AlternatingGroupGraph[n]],{n,9}])//Timing

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

Table[Piecewise[{{0,n==1}},n!(n-2)/2],{n,20}]

seq=Join[{0},Table[n!(n-2)/2,{n,2,50}]]

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

Table[n!(n-2)/2,{n,0,10}]

GeneratingFunction[n!(n-2)/2,n,x]

ExponentialGeneratingFunction[Piecewise[{{0,n<3}},n!(n-2)/2],n,x]

SeriesCoefficient[((x^3)/(2 (((-1+x))^2))),{x,0,n}]

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

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

CoefficientList[Series[((x^2)/(2 (((-1+x))^2))),{x,0,19}],x]Range[20]!

FullSimplify[(1-x)Exp[-x]+(x^2)(CoshIntegral[x]-SinhIntegral[x])-1+2x-((1/2) (3+2 EulerGamma+2 Log[x]) (x^2))]

Series[(1-x)Exp[-x]+(x^2)(CoshIntegral[x]-SinhIntegral[x])-1+2x-((1/2) (3+2 EulerGamma+2 Log[x]) (x^2)),{x,0,20}]

FullSimplify[SeriesCoefficient[(1-x)Exp[-x]+(x^2)(CoshIntegral[x]-SinhIntegral[x])-1+2x-((1/2) (3+2 EulerGamma+2 Log[x]) (x^2)),{x,0,n}],n∈Integers&&n>=3]

EdgeCoverCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"EdgeCoverCount"],{n,2,6}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n],n>0&&n∈Integers]

coef=FindLinearRecurrence[DeleteMissing@seq]

Length@coef

Take[seq,6]

LinearRecurrence[{11,-24,-21,33,34,8},{2,34,341,2902,23092,178393},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

CoefficientList[Series[((-2-12 x-15 (x^2)-9 (x^3)-2 (x^4))/(-1+11 x-24 (x^2)-21 (x^3)+33 (x^4)+34 (x^5)+8 (x^6))),{x,0,20}],x]

EdgeCoverNumber

A001710

seq=Table[GraphData[{"AlternatingGroupGraph",n},"EdgeCoverNumber"],{n,2,6}]

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

Range[6]!/2

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"AlternatingGroupGraph",n},"EdgeCoverPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]//Column

FindLinearRecurrence[DeleteMissing@Drop[poly,1]]//Head

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

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

FullSimplify[%-poly]

EulerianCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"EulerianCycleCount"],{n,2,6}]

FaceCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"FaceCount"],{n,2,6}]

FindSequenceFunction[seq,n]

FlowPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"FlowPolynomial"],{n,k0=1,kmax=6}]/.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]

FractionalChromaticNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"FractionalChromaticNumber"],{n,2,6}]

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

FractionalCliqueNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"FractionalCliqueNumber"],{n,2,6}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"FractionalEdgeChromaticNumber"],{n,2,6}]

Girth

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Girth"],{n,2,6}]

HamiltonDecompositionCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HamiltonDecompositionCount"],{n,2,6}]

Hamiltonian

Table[GraphData[{"AlternatingGroupGraph",n},"Hamiltonian"],{n,2,6}]

HamiltonianCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HamiltonianCycleCount"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

HamiltonianNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HamiltonianNumber"],{n,2,6}]

FindSequenceFunction[seq,n]

HamiltonianPathCount

A317485

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HamiltonianPathCount"],{n,2,6}]

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

HamiltonianWalkCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HamiltonianWalkCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

HararyIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HararyIndex"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

HexagonCount

A317486

seq=Table[GraphData[{"AlternatingGroupGraph",n},"HexagonCount"],{n,2,6}]

(seq=Table[CycleCount[AlternatingGroupGraph[n],6],{n,8}])//Timing

(seq=Table[CycleCount[AlternatingGroupGraph[n],6],{n,9}])//Timing

FindLinearRecurrence[Drop[seq,2]]

FindGeneratingFunction[seq,x]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"IdiosyncraticPolynomial"],{n,k0=1,kmax=6}]/.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]

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

FullSimplify[%-poly]

IndependenceNumber

A001710

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IndependenceNumber"],{n,2,6}]

seq=Table[GraphData[{"AlternatingGroupGraph",n},"V"],{n,2,6}]

Table[Piecewise[{{1,n==2||n==3}},n!/6],{n,2,10}]

Do[Print[{{"AlternatingGroupGraph",n},MemoryConstrained[First@FindIndependentVertexSet[AlternatingGroupGraph[n]],100*^9]//Timing}],{n,7}]

From Stan Wagon, Aug 8, 2018

Table 1.  Known results for the independence number of (Subscript[A, n+k,n]). Perfect strategies exist in all cases shown except the red entries, and perhaps the bold one.

{1,1,4,20,120,840,6720}

IndependencePolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"IndependencePolynomial"],{n,k0=1,kmax=6}]/.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}]

Simplify[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]]

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

CoefficientList[Table[(2^(-n)) ((((1-(Sqrt[1+4 x])))^n)+(((1+(Sqrt[1+4 x])))^n)),{n,10}]//Expand,x]//Column

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

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

Subtract@@@Table[{x(x+2)+(x^(n/2))LucasL[n,1/(Sqrt[x])],GraphData[{"AlternatingGroupGraph",n},"IndependencePolynomial"][x]},{n,3,10}]//Expand

IndependenceRatio

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IndependenceRatio"],{n,2,6}]

From Stan Wagon 2018-07-30: "FWIW Piotr has been expessing doubts that the 1/3 pattern continues. I am still hopeful as it is too beautiful."

FullSimplify[FindSequenceFunction[Rest@seq,n-3],n>3&&n∈Integers]

Table[Floor[2 n/3]/(2n),{n,2,6}]

n!/2

IndependentEdgeSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IndependentEdgeSetCount"],{n,2,6}]

IndependentVertexSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IndependentVertexSetCount"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimplify

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

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

Table[LucasL[n]+3,{n,2,6}]

IntersectionNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IntersectionNumber"],{n,2,6}]

IrredundancePolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"IrredundancePolynomial"],{n,k0=4,20}]/.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}]

Simplify[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]

IrredundantSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"IrredundantSetCount"],{n,2,6}]

coef=FindLinearRecurrence[Rest@seq]

Length[coef]

Take[seq,1+{1,8}]

Join[{7},LinearRecurrence[{3,-2,-1,2,-2,0,2,-1},{22,29,39,60,94,151,241,400},20]]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[Drop[seq,2],n-4]

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

CoefficientList[Series[((7+x-23 (x^2)+3 (x^3)+9 (x^4)-9 (x^5)+14 (x^6)+2 (x^7)-6 (x^8))/((((-1+x))^2) (1-x-(x^2)-(x^4)+(x^6)))),{x,0,20}],x]

Refine[SeriesCoefficient[gf,{x,0,n}],n>3]

terms=List@@res;

N[terms/.n->0]//Chop

terms[[All,1]]

Expand[1+2 (1+n)]

Table[3+2 n+RootSum[1-((#1)^2)-((#1)^4)-((#1)^5)+((#1)^6)&,((#)^n)&],{n,40}]

Table[Piecewise[{{7,n==3}},3+2 n+RootSum[1-((#1)^2)-((#1)^4)-((#1)^5)+((#1)^6)&,((#)^n)&]],{n,2,6}]

KirchhoffIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"KirchhoffIndex"],{n,2,6}]

(seq=Table[KirchhoffIndex@AlternatingGroupalGraph[n],{n,3,40}])//Timing

FindSequenceFunction[Drop[seq,3],n-5]//Head//Timing

FindLinearRecurrence[Drop[seq,3]]//Head//Timing

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

KirchhoffSumIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"KirchhoffSumIndex"],{n,2,6}]

(seq=Table[KirchhoffSumIndex@AlternatingGroupalGraph[n],{n,3,40}])//Timing

FindSequenceFunction[Drop[seq,3],n-5]//Head//Timing

FindLinearRecurrence[Drop[seq,3]]//Head//Timing

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

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"LaplacianPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

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

FullSimplify[%-poly]

LeafCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"LeafCount"],{n,2,6}]

Likelihood

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Likelihood"],{n,2,6}]

LongestCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"LongestCycleCount"],{n,2,6}]

LongestPathLength

seq=Table[GraphData[{"AlternatingGroupGraph",n},"LongestPathLength"],{n,2,6}]

LongestPathCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"LongestPathCount"],{n,2,6}]

LovaszNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"LovaszNumber"],{n,2,6}]

MatchingGeneratingPolynomial

Factor@(poly=Table[GraphData[{"AlternatingGroupGraph",n},"MatchingGeneratingPolynomial"],{n,k0=1,kmax=6}]/.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]]

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

Table[(1/(((1+4 x))^(3/2)))(2^(-n)) (-n x (2+7 x) ((((1-(Sqrt[1+4 x])))^n)-(((1+(Sqrt[1+4 x])))^n))+(n^2) (x^2) (Sqrt[1+4 x]) ((((1-(Sqrt[1+4 x])))^n)+(((1+(Sqrt[1+4 x])))^n))+(((1+4 x))^(3/2)) ((((1-(Sqrt[1+4 x])))^n)+(((1+(Sqrt[1+4 x])))^n))),{n,5}]//Factor

Table[(1/(((1+4 x))^(3/2)))(2^(-n)) (-n x (2+7 x) ((((1-(Sqrt[1+4 x])))^n)-(((1+(Sqrt[1+4 x])))^n))+((((1+4 x))^(3/2))+(n^2) (x^2) (Sqrt[1+4 x]) )((((1-(Sqrt[1+4 x])))^n)+(((1+(Sqrt[1+4 x])))^n))),{n,5}]//Factor

MatchingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MatchingNumber"],{n,2,6}]

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

MatchingPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"MatchingPolynomial"],{n,k0=1,kmax=6}]/.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]]

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

MaximalCliqueCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximalCliqueCount"],{n,2,6}]

FindSequenceFunction[Rest@seq,n]

MaximalIndependencePolynomial

(poly=Table[GraphData[{"AlternatingGroupGraph",n},"MaximalIndependencePolynomial"],{n,2,6}]/.f_Function:>f[x])//Column

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximalIndependentEdgeSetCount"],{n,2,6}]

(seq=Table[Length@FindAllMaximalIndependentEdgeSets[AlternatingGroupalGraph[n]],{n,3,35}])//Timing

coef=FindLinearRecurrence[seq]

Length[coef]

LinearRecurrence[coef,Take[seq,Length@coef],{-1,20}]//Take[#,13]&

seq=LinearRecurrence[{0,5,3,-10,-12,7,18,4,-11,-8,1,3,1},{2,4,12,14,40,56,112,178,306,482,792,1214,1924},40]

FindSequenceFunction[Drop[seq,2],n-4]//Head//Timing

gf=x FindGeneratingFunction[seq,x,ValidationLength->5]

CoefficientList[Series[-((2 (1+2 x+(x^2)-6 (x^3)-6 (x^4)+7 (x^5)+12 (x^6)-(x^7)-9 (x^8)-6 (x^9)+2 (x^11)+(x^12)))/((((-1+(x^2)))^2) (((-1+(x^2)+(x^3)))^3))),{x,0,20}],x]

Refine[SeriesCoefficient[gf,{x,0,n}],n>0&&n∈Integers]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximalIndependentVertexSetCount"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimpify

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

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

MaximalIrredundantSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximalIrredundantSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-4]//Head//Timing

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

FindGeneratingFunction[seq,x,ValidationLength->5]//Head//Timing

MaximumCliqueCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumCliqueCount"],{n,2,6}]

FindSequenceFunction[Rest@seq,n]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumIndependentEdgeSetCount"],{n,2,6}]

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

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

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumIndependentVertexSetCount"],{n,2,6}]

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

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

MaximumIrredundantSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumIrredundantSetCount"],{n,2,6}]

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

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

MaximumLeafNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumLeafNumber"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-3]

MaximumVertexDegree

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MaximumVertexDegree"],{n,2,6}]

MeanDistance

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MeanDistance"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

MinimalDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimalDominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-3]

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

Refine[SeriesCoefficient[gf,{x,0,n}],n>3]

MinimalDominatingSetSignature

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimalDominatingSetSignature"],{n,2,6}]//Column

FindSequenceFunction[DeleteMissing@seq,n]//Factor

MinimalEdgeCoverCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimalEdgeCoverCount"],{n,2,6}]

seq={1,6,24,74,180,464,1113,2646,6360,15222,36795,89584,219635,542320,1346881,3361998,8427172,21195416,53455740,135112332,342093443,867325032,2201286622,5591469852,14211796995,36139507614,91934054637,233934039872,595393224041,1515602413390};

seq=Rest@CoefficientList[Series[x*(1+x)*(1+x-2*x^2-14*x^3-39*x^4+63*x^5+69*x^6+55*x^7-39*x^8-85*x^9-118*x^10-102*x^11-63*x^12-27*x^13-7*x^14-x^15)/((1-x^2-x^3)^3*(1-x-x^2-x^3)^2*(1-2*x-x^2-x^3)),{x,0,50}],x]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,18]

LinearRecurrence[{4,1,-12,-15,24,49,6,-73,-76,5,80,72,14,-30,-34,-19,-6,-1},{1,6,24,74,180,464,1113,2646,6360,15222,36795,89584,219635,542320,1346881,3361998,8427172,21195416},30]

FindSequenceFunction[seq,n]

gf=x Factor@FindGeneratingFunction[seq,x,ValidationLength->5]

CoefficientList[Series[-(((1+x) (-1-x+2 (x^2)+14 (x^3)+39 (x^4)-63 (x^5)-69 (x^6)-55 (x^7)+39 (x^8)+85 (x^9)+118 (x^10)+102 (x^11)+63 (x^12)+27 (x^13)+7 (x^14)+(x^15)))/((((-1+(x^2)+(x^3)))^3) (((-1+x+(x^2)+(x^3)))^2) (-1+2 x+(x^2)+(x^3)))),{x,0,20}],x]

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

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>0&&n∈Integers]

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimalTotalDominatingSetCount"],{n,2,6}]

seq={2,4,9,12,15,21,21,20,30,45,44,49,65,77,98,132,153,180,247,329,409,528,690,889,1180,157637,2657,3538,4684,6169,8164,10783,14229,18877,25036,33078,43757,57996,76809,101721,134773,178450,236284,313097,414828,549383};

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length[coef]]

LinearRecurrence[{2,-1,1,-1,0,0,-1,0,1,1,-1},{2,4,9,12,15,21,21,20,30,45,44},20]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x,ValidationLength->5]

CoefficientList[Series[((2+3 (x^2)-4 (x^3)-2 (x^4)-2 (x^5)-9 (x^6)-2 (x^7)+9 (x^8)+12 (x^9)-9 (x^10))/((((-1+x))^2) (1-(x^3)-(x^4)-(x^5)-(x^6)+(x^8)+(x^9)))),{x,0,20}],x]

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

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

Table[(Root[1-#1+((#1)^3)&,2] (((Root[-1+((#1)^2)+((#1)^3)&,1])^((2+n)/2))-((Root[-1+((#1)^2)+((#1)^3)&,2])^((2+n)/2)))+Root[1-#1+((#1)^3)&,3] (((Root[-1+((#1)^2)+((#1)^3)&,1])^((2+n)/2))-((Root[-1+((#1)^2)+((#1)^3)&,3])^((2+n)/2)))),{n,2,30,2}]//RootReduce

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

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

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumCoveringsByMaximalCliquesCount"],{n,2,6}]

seq={1,4,30,12,98,28,270,60,682,124,1638,252,3810,508,8670,1020,19418,2044,42966,4092,94162,8188,204750,16380,442314,32764,950214,65532,2031554,131068,4325310,262140,9174970,524284,19398582,1048572,40894386,2097148,85983150};

Join[{1},Table[If[Mod[n,2]==0,2,n](2^Ceiling[n/2]-2),{n,4,20}]]

Table[Piecewise[{{1,n==3},{2(2^Ceiling[n/2]-2),Mod[n,2]==0},{(2^Ceiling[n/2]-2)n,Mod[n,2]==1}}],{n,2,6}]

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-3]//Head//Timing

coef=FindLinearRecurrence[DeleteMissing@Drop[seq,1]]

Length@coef

Take[seq,1+{1,8}]

Join[{1},LinearRecurrence[{0,6,0,-13,0,12,0,-4},{4,30,12,98,28,270,60,682},20]]

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

CoefficientList[Series[((1+4 x+24 (x^2)-12 (x^3)-69 (x^4)+8 (x^5)+60 (x^6)-20 (x^8))/(((1-3 (x^2)+2 (x^4)))^2)),{x,0,20}],x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>3&&n∈Integers]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumConnectedDominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-4]//FullSimplify

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumDistinguishingLabelingCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteCases[seq,_Missing],x]

MinimumDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumDominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Drop[seq,4],n-6]//Factor

MinimumEdgeCoverCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumEdgeCoverCount"],{n,2,6}]

seq={1,2,21,8,85,18,217,32,441,50,781,72,1261,98,1905,128,2737,162,3781,200,5061,242,6601,288,8425,338,10557,392,13021,450,15841,512,19041,578,22645,648,26677,722,31161,800,36121,882,41581,968,47565,1058,54097,1152};

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

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,8]

LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,2,21,8,85,18,217,32},20]

Table[Floor[(n+3)/2],{n,20}]

gf=x FindGeneratingFunction[seq,x]

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

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

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

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumTotalDominatingSetCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@Drop[seq,2],n-4]

MinimumVertexCoverCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumVertexCoverCount"],{n,2,6}]

MinimumVertexDegree

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumVertexDegree"],{n,2,6}]

MolecularTopologicalIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MolecularTopologicalIndex"],{n,2,6}]

FindLinearRecurrence[seq

FindSequenceFunction[seq,n]//Factor

Nonhamiltonian

Table[GraphData[{"AlternatingGroupGraph",n},"Nonhamiltonian"],{n,2,6}]

OddChordlessCycleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"OddChordlessCycleCount"],{n,2,6}]

PathCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"PathCount"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

PathPolynomial

(poly=Factor[Table[GraphData[{"AlternatingGroupGraph",n},"PathPolynomial"],{n,k0=1,kmax=6}]/.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[{"AlternatingGroupGraph",n},"PentagonCount"],{n,2,6}]

(seq=Table[CycleCount[AlternatingGroupGraph[n],5],{n,8}])//Timing

(seq=Table[CycleCount[AlternatingGroupGraph[n],5],{n,9}])//Timing

Planar

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Planar"],{n,2,6}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ProjectivePlaneCrossingNumber"],{n,2,6}]

Radius

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Radius"],{n,2,6}]

RankPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"RankPolynomial"],{n,k0=1,kmax=6}]/.f_Function:>f[x,y])

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}]

Simplify[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]]

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

RectilinearCrossingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"RectilinearCrossingNumber"],{n,2,6}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"ReliabilityPolynomial"],{n,k0=1,kmax=6}]/.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]

FullSimplify[poly2-poly]

ShannonCapacity

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ShannonCapacity"],{n,2,6}]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"SigmaPolynomial"],{n,k0=1,kmax=6}]/.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]

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

FullSimplify[%-poly]

Skewness

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Skewness"],{n,2,6}]

SpanningTreeCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"SpanningTreeCount"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimplify

SquareCount

A317487

seq=Table[GraphData[{"AlternatingGroupGraph",n},"SquareCount"],{n,2,6}]

(seq=Table[CycleCount[AlternatingGroupGraph[n],4],{n,9}])//Timing

(seq=Table[CycleCount[AlternatingGroupGraph[n],4],{n,10}])//Timing

FindLinearRecurrence[Drop[seq,3]]

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

FindGeneratingFunction[seq,x]

StabilityIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"StabilityIndex"],{n,2,6}]

TopologicalIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"TopologicalIndex"],{n,2,6}]

ToroidalCrossingNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"ToroidalCrossingNumber"],{n,2,6}]

TotalDominatingSetCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"TotalDominatingSetCount"],{n,2,6}]

FullSimplify[FindSequenceFunction[DeleteMissing@seq,n],n∈Integers]

TotalDominationNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"TotalDominationNumber"],{n,2,6}]

TotalDominationPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"TotalDominationPolynomial"],{n,k0=1,kmax=6}]/.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]

Traceable

Table[GraphData[{"AlternatingGroupGraph",n},"Traceable"],{n,2,6}]

Triameter

seq=Table[GraphData[{"AlternatingGroupGraph",n},"Triameter"],{n,2,6}]

FindSequenceFunction[DeleteMissing@seq,n]

TriangleCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"TriangleCount"],{n,2,6}]

(seq=Table[CycleCount[AlternatingGroupGraph[n],3],{n,8}])//Timing

(seq=Table[CycleCount[AlternatingGroupGraph[n],3],{n,9}])//Timing

TuttePolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"TuttePolynomial"],{n,k0=1,kmax=6}]/.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}]

Refine[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]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"AlternatingGroupGraph",n},"Untraceable"],{n,2,6}]

VertexConnectivity

Table[GraphData[{"AlternatingGroupGraph",n},"VertexConnectivity"],{n,2,6}]

VertexCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"V"],{n,2,6}]

FindSequenceFunction[seq,n-1]//FullSimplify

FunctionExpand[%]

Table[n!/2,{n,2,6}]

VertexCoverCount

seq=Table[GraphData[{"AlternatingGroupGraph",n},"VertexCoverCount"],{n,2,6}]

FindSequenceFunction[seq,n]

VertexCoverNumber

seq=Table[GraphData[{"AlternatingGroupGraph",n},"VertexCoverNumber"],{n,2,6}]

FindSequenceFunction[seq,n]//FullSimplify

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

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>3]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"AlternatingGroupGraph",n},"VertexCoverPolynomial"],{n,k0=1,kmax=6}]/.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]]

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

poly2=Table[LinearRecurrence[{x^2,2 x^3,x^4},{x^2,x^3 (4+x),x^4 (3+6 x+x^2)},{n}][[1]],{n,k0,20}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"AlternatingGroupGraph",n},"VertexDegrees"],{n,3,10}]

WienerIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"WienerIndex"],{n,2,6}]

FindSequenceFunction[seq,n]//Factor

WienerSumIndex

seq=Table[GraphData[{"AlternatingGroupGraph",n},"WienerSumIndex"],{n,2,6}]

(seq=Table[WienerSumIndex@AlternatingGroupalGraph[n],{n,3,40}])//Timing

FindSequenceFunction[Drop[seq,3],n-5]//Head//Timing

FindLinearRecurrence[Drop[seq,3]]//Head//Timing

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

Recognize

Standard Generators

gs=Table[UndirectedGraph[CayleyGraph[System`AlternatingGroup[n]]],{n,7}];//Timing

VertexCount/@gs

DeleteCases[rg=MapIndexed[RecognizeGraph[#1]->#2[[1]]&,gs],{}->_]

Cases[rg,HoldPattern[{}->_]]

Table[VertexCount[AlternatingGroupGraph2[n]],{n,3,7}]

Table[Max[VertexDegree[AlternatingGroupGraph2[n]]],{n,3,6}]

Table[AlternatingGroupGraph2[n],"Diameter"],{n,3,6}]

Jwo

RecognizeGraph/@Table[AlternatingGroupGraph[n],{n,3,7}]

VertexCount/@Table[AlternatingGroupGraph[n],{n,3,7}]

Table[n!/2,{n,3,7}]

Max[VertexDegree[#]]&/@Table[AlternatingGroupGraph[n],{n,3,6}]

Table[2(n-2),{n,3,6}]

GraphDiameter/@Table[AlternatingGroupGraph[n],{n,3,7}]

Table[Floor[3(n-2)/2],{n,3,7}]

Graphs from different sets of Generators

(Subscript[A, 3])

GroupGenerators[System`AlternatingGroup[3]]

elem=Rest[GroupElements[System`AlternatingGroup[3]]]

gs=PermutationGroup/@Rest[Subsets[elem]]

Union[RecognizeGraph/@(UndirectedGraph[CayleyGraph[#]]&/@gs)]

(Subscript[A, 4])

GroupGenerators[System`AlternatingGroup[4]]

Length[elem=Rest[GroupElements[System`AlternatingGroup[4]]]]

Length[gs=PermutationGroup/@Rest[Subsets[elem]]]

Length[gs2=Select[gs,GroupOrder[#]===GroupOrder[System`AlternatingGroup[4]]&]]

(graphs=Union[RecognizeGraph/@(UndirectedGraph[CayleyGraph[#]]&/@gs)])//Timing

Select[graphs,GraphData[#,"Disconnected"]&]

Hamiltonian Cycles

http://link.springer.com/article/10.1007/s11227-011-0614-4

claims

“we will show that any alternating group graph (Subscript[AG, n]) , where n≥3 is an integer, contains 2n−4 mutually independent Hamiltonian cycles.”

But this seems not to be the case :(

Table[2n-4,{n,3,5}]

3

OK for triangle graph:  {{1->2, 2->3, 3->1}, {1->3, 3->2, 2->1}}

4

Length[uc=GetGraphData[{"AlternatingGroupGraph",4},"UndirectedHamiltonianCycles"]]

Length[uc=FindHamiltonianCycle[GraphData[{"AlternatingGroupGraph",4}],All][[All,All,1]]]//Timing

2 4-4

Length[dc=DirectedCycles[uc]]

Using Edges

Length[de=Partition[#,2,1,1]&/@dc]

ide=Union[de,SameTest->(Intersection[#1,#2]=!={}&)]

ide[[All,All,1]]

Length[%]

only 3...

Using Vertices (what paper claims)

Length[dc]

dc[[All,1]]//Union

DigitsOverlapQ[d1_,d2_]:=(Or@@Equal@@@Transpose[{d1,d2}])

Union[Rest/@dc,SameTest->(DigitsOverlapQ[#1,#2]&)]

only 3...

Testing

6

g=AlternatingGroupGraph[6]

autog=AutomorphismGroup[g];

GroupOrder[autog]

PermutationList[#,720]&/@GroupElements[autog]

MaxMemoryUsed[]/10^6.

(5040/720)^2

7

g=AlternatingGroupGraph[7]

autog=AutomorphismGroup[g];

GroupOrder[autog]

PermutationList[#,5040]&/@GroupElements[autog]

5040^2

GraphData

file=OpenAppend["~/alternating.m"]

Do[WriteString[file,
GraphDataString[
AlternatingGroupGraph[n],
"StandardName"->{"AlternatingGroupGraph",n},
"Name"->ToString[n]<>"-alternating group graph",
"Vertices"->None,
"Information"->"AlternatingGroupGraph",
"Skip"->{"ChromaticPolynomial","ClawFree","Line"},
"Classes"->{"AlternatingGroupGraph","CayleyGraph"},
"NotationRules"->{"AlternatingGroupGraph"->n},
TimeConstraint->10,
Debug->True]],{n,7,7}]

Close[file]