Sierpiński Gasket Graph

Author

Eric W. Weisstein
April 11, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

References

https://mathematica.stackexchange.com/questions/30728/sierpinski-carpet-with-graphdata

Figures

GraphicsRow[Table[SierpinskiGasketGraph[n,VertexSize->{"Scaled",.05},VertexStyle->Red,EdgeStyle->Black,PlotLabel->Style[Subscript[S,n],Directive[Black,FontFamily->"Times"]]],{n,4}],ImageSize->500,Dividers->All]

Coloring

<<MathWorld`Graphs`

Uniquely 3-colorable up to permutation of colors

2

Length[colorings2=Select[Tuples[Range[3],6],VertexColoringQ[SierpinskiGasketGraph[2],#]&]]//Timing

gr1=GraphicsGrid[Partition[VertexColoredGraph[SierpinskiGasketGraph[2],#,VertexSize->.1]&/@colorings2,UpTo[3]]]

3

Length[colorings3=With[{g=SierpinskiGasketGraph[3]},Select[Tuples[Range[3],VertexCount[g]],VertexColoringQ[g,#]&]]]//Timing

colorings3

MaxMemoryUsed[]/10^9.GB

gr2=GraphicsGrid[Partition[VertexColoredGraph[SierpinskiGasketGraph[3],#,VertexSize->.2]&/@colorings3,UpTo[3]]]

4

Length[colorings3=With[{g=SierpinskiGasketGraph[4]},Select[Tuples[Range[3],VertexCount[g]],VertexColoringQ[g,#]&]]]//Timing

Graphic

GraphicsRow[{gr1,gr2},Spacings->-50,Alignment->Center,ImageSize->700]

Construction

By hand

SierpinskiMesh[1]

v=MeshCoordinates[SierpinskiMesh[1]]

c0=MeshCells[SierpinskiMesh[1],0]

c1=MeshCells[SierpinskiMesh[1],1]

c1[[All,1]]

Graphics[{Point[v],GraphicsComplex[v,Line[c1[[All,1]]]],MapIndexed[Text[#2[[1]],#+.02]&,v]}]

RecognizeGraph/@Table[SierpinskiGasketGraph[n],{n,5}]

MeshConnectivityGraph (V12.1)

MeshConnectivityGraph@SierpinskiMesh[3]

RecognizeGraph[%]

IGraph

Needs["IGraphM`"]

IGMeshGraph@SierpinskiMesh[3]

RecognizeGraph[%]

Special Cases

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

Properties

Acyclic

seq=Table[GraphData[{"SierpinskiSieve",n},"Acyclic"],{n,5}]

AdjacencyMatrixCount

seq=Table[GraphData[{"SierpinskiSieve",n},"AdjacencyMatrixCount"],{n,5}]

FullSimplify[((((1/2) (3+(3^n)))!)/6)]

Table[((((1/2) (3+(3^n)))!)/6),{n,5}]==seq

Arboricity

Probably 2

seq=Table[GraphData[{"SierpinskiSieve",n},"Arboricity"],{n,5}]

ArcTransitive

seq=Table[GraphData[{"SierpinskiSieve",n},"ArcTransitive"],{n,5}]

ArcTransitivity

seq=Table[GraphData[{"SierpinskiSieve",n},"ArcTransitivity"],{n,5}]

AutomorphismCount

seq=Table[GraphData[{"SierpinskiSieve",n},"AutomorphismCount"],{n,5}]

BalabanIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"BalabanIndex"],{n,4}]

Bandwidth

seq=Table[GraphData[{"SierpinskiSieve",n},"Bandwidth"],{n,5}]

BridgeCount

seq=Table[GraphData[{"SierpinskiSieve",n},"BridgeCount"],{n,5}]

BurningNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"BurningNumber"],{n,5}]

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"CharacteristicPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Chordal

seq=Table[GraphData[{"SierpinskiSieve",n},"Chordal"],{n,5}]

ChordCount

seq=Table[GraphData[{"SierpinskiSieve",n},"ChordCount"],{n,5}]

GraphData[{"SierpinskiSieve",2},"Chords"]

HighlightGraph[GraphData[{"SierpinskiSieve",2},"Graph","Labeled"],GraphData[{"SierpinskiSieve",2},"Chords"]]

HighlightGraph[GraphData[{"SierpinskiSieve",3},"Graph","Labeled"],GraphData[{"SierpinskiSieve",3},"Chords"]]

GraphData[{"SierpinskiSieve",4},"Chords"]

GraphData[{"SierpinskiSieve",4},"Graph","Labeled"]

Chordless

seq=Table[GraphData[{"SierpinskiSieve",n},"Chordless"],{n,5}]

ChordlessCycleCount

seq=Table[GraphData[{"SierpinskiSieve",n},"ChordlessCycleCount"],{n,5}]

FindSequenceFunction[seq,n]

Chords

seq=Table[GraphData[{"SierpinskiSieve",n},"Chords"],{n,5}]

ChromaticallyUnique

seq=Table[GraphData[{"SierpinskiSieve",n},"ChromaticallyUnique"],{n,5}]

GraphData[{"ChromaticallyUnique","SierpinskiSieve"}]

ChromaticInvariant

seq=Table[GraphData[{"SierpinskiSieve",n},"ChromaticInvariant"],{n,5}]

FindSequenceFunction[DeleteMissing@seq,n]

ChromaticNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"ChromaticNumber"],{n,5}]

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"ChromaticPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Circumference

seq=Table[GraphData[{"SierpinskiSieve",n},"Circumference"],{n,5}]

FindSequenceFunction[seq,n]

Classes

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

CliqueCount

seq=Table[GraphData[{"SierpinskiSieve",n},"CliqueCount"],{n,5}]

(seq=Table[Length@FindAllCliques[SierpinskiGasketGraph[n]],{n,7}])//Timing

FullSimplify[1+(1/2) (3+(3^n))x+(3^n)(x^2)+Piecewise[{{1,n==1}},4 (3^(-2+n))](x^3)/.x->1]

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

CliqueCoveringNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"CliqueCoveringNumber"],{n,5}]

(seq=Table[CliqueCoveringNumber[SierpinskiGasketGraph[n]],{n,4}])//Timing

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

CliqueNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"CliqueNumber"],{n,5}]

CliquePolynomial

(poly=Expand[Table[GraphData[{"SierpinskiSieve",n},"CliquePolynomial"],{n,k0=1,kmax=5}]/.f_Function:>f[x]])//Column

(poly=Table[CliquePolynomial[SierpinskiGasketGraph[n],x],{n,6}])//Column//Timing

FindSequenceFunction[#,n]&/@Transpose[CoefficientList[poly,x]]

FindSequenceFunction[{4,12,36,108,324},n-1]

Table[1+(1/2) (3+(3^n))x+(3^n)(x^2)+Piecewise[{{1,n==1}},4 (3^(-2+n))](x^3),{n,6}]//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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

CochromaticGraphNames

Table[GraphData[{"SierpinskiSieve",n},"CochromaticGraphNames"],{n,5}]//Column

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

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"SierpinskiSieve",n},"ComplementOddChordlessCycleCount"],{n,5}]

FindSequenceFunction[Take[seq,{3,5}],n]

ConnectedComponentCount

seq=Table[GraphData[{"SierpinskiSieve",n},"ConnectedComponentCount"],{n,5}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"ConnectedDominatingSetCount"],{n,5}]

FindSequenceFunction[seq,n]

ConnectedDominationNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"ConnectedDominationNumber"],{n,5}]

FindSequenceFunction[seq,n]

ConnectedDominationPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"ConnectedDominationPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

A292067

seq=Table[GraphData[{"SierpinskiSieve",n},"ConnectedInducedSubgraphCount"],{n,5}]

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"CharacteristicPolynomial"],{n,k0=1,kmax=5}]/.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,6}]

FullSimplify[poly2-poly]

CrossingNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"CrossingNumber"],{n,5}]

CycleCount

A234634

seq=Table[GraphData[{"SierpinskiSieve",n},"CycleCount"],{n,5}]

seq=Import["https://oeis.org/A234634/b234634.txt","Table"][[All,2]]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

CyclePolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"CyclePolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"SierpinskiSieve",n},"Cyclic"],{n,5}]

CyclomaticNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"CyclomaticNumber"],{n,5}]

FindSequenceFunction[seq,n]

DetourIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"DetourIndex"],{n,5}]

FindSequenceFunction[DeleteMissing@seq,n]

DetourMatrix

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

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

DetourPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"DetourPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Diameter

seq=Table[GraphData[{"SierpinskiSieve",n},"Diameter"],{n,5}]

FindSequenceFunction[seq,n]

DistanceMatrix

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

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

DistancePolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"DistancePolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

DistinguishingNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"DistinguishingNumber"],{n,5}]

DomaticNumber

Possibly 3?

seq=Table[GraphData[{"SierpinskiSieve",n},"DomaticNumber"],{n,5}]

DominatingSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"DominatingSetCount"],{n,5}]

DominationNumber

(3^(n-2)) for n >= 3

Teguia, A.~M. and Godbole, A.~P.  "Sierpńiski Gasket Graphs and Some of Their Properties."  Australasian J. Combin. 35, 181–192, 2006.

seq=Table[GraphData[{"SierpinskiSieve",n},"DominationNumber"],{n,5}]

(seq=Table[DominationNumber[SierpinskiGasketGraph[n]],{n,5}])//Timing

DominationPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"DominationPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

EdgeChromaticNumber

Probably 4

seq=Table[GraphData[{"SierpinskiSieve",n},"EdgeChromaticNumber"],{n,5}]

FindSequenceFunction[seq,n]-2

gf=x FindGeneratingFunction[seq,x]

EdgeConnectivity

Table[GraphData[{"SierpinskiSieve",n},"EdgeConnectivity"],{n,5}]

EdgeCount

seq=Table[GraphData[{"SierpinskiSieve",n},"EdgeCount"],{n,5}]

FindSequenceFunction[seq,n]

EdgeCoverCount

seq=Table[GraphData[{"SierpinskiSieve",n},"EdgeCoverCount"],{n,5}]

EdgeCoverNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"EdgeCoverNumber"],{n,5}]

FindSequenceFunction[seq,n]

x FindGeneratingFunction[seq,x]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"EdgeCoverPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"SierpinskiSieve",n},"EulerianCycleCount"],{n,5}]

FaceCount

seq=Table[GraphData[{"SierpinskiSieve",n},"FaceCount"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

FlowPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"FlowPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

FractionalChromaticNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"FractionalChromaticNumber"],{n,5}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"FractionalEdgeChromaticNumber"],{n,5}]

Girth

seq=Table[GraphData[{"SierpinskiSieve",n},"Girth"],{n,5}]

Gonality

seq=Table[GraphData[{"SierpinskiSieve",n},"Gonality"],{n,5}]

HamiltonConnected

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonConnected"],{n,5}]

HamiltonDecompositionCount

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonDecompositionCount"],{n,5}]

Hamiltonian

Table[GraphData[{"SierpinskiSieve",n},"Hamiltonian"],{n,5}]

HamiltonianCycleCount

A246959

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonianCycleCount"],{n,5}]

seq=Table[Piecewise[{{1,n==1||n==2}},8 12^((3^(n-2)-3)/2)],{n,20}];

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

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

gf=x FindGeneratingFunction[seq,x]

Join[{1,1},RecurrenceTable[{a[3]==8,a[n]==(3a[n-1])^3},a,{n,3,8}]]

HamiltonianNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonianNumber"],{n,5}]

FindSequenceFunction[seq,n]

HamiltonianPathCount

A246957

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonianPathCount"],{n,5}]

Explicit formula and asymptotic are given by Chang and Chen (2011).

S.-C. Chang, L.-C. Chen. Hamiltonian walks on the Sierpinski gasket, J. Math. Phys. 52 (2011), 023301. doi:10.1063/1.3545358. arXiv:0909.5541

HamiltonianWalkCount

Probably 1

seq=Table[GraphData[{"SierpinskiSieve",n},"HamiltonianWalkCount"],{n,5}]

HararyIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"HararyIndex"],{n,5}]

(seq=Table[HararyIndex[SierpinskiGasketGraph[n]],{n,8}])//Timing

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

HexagonCount

seq=Table[GraphData[{"SierpinskiSieve",n},"HexagonCount"],{n,5}]

FindSequenceFunction[seq,n]

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

gf=x FindGeneratingFunction[seq,x]

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

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"IdiosyncraticPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

IndependenceNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"IndependenceNumber"],{n,5}]

(seq=Table[IndependenceNumber[SierpinskiGasketGraph[n]],{n,5}])//Timing

(seq=Table[IndependenceNumber[SierpinskiGasketGraph[n]],{n,6}])//Timing

FindSequenceFunction[seq,n]

IndependencePolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"IndependencePolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"SierpinskiSieve",n},"IndependenceRatio"],{n,5}]

FindSequenceFunction[seq,n]

IndependentEdgeSetCount

A292968

seq=Table[GraphData[{"SierpinskiSieve",n},"IndependentEdgeSetCount"],{n,5}]

FindSequenceFunction[DeleteMissing@seq,n]

IndependentVertexSetCount

A292708

seq=Table[GraphData[{"SierpinskiSieve",n},"IndependentVertexSetCount"],{n,5}]

FindSequenceFunction[seq,n]

IntersectionNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"IntersectionNumber"],{n,5}]

IrredundantSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"IrredundantSetCount"],{n,5}]

KirchhoffIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"KirchhoffIndex"],{n,5}]

(seq=Table[KirchhoffIndex[SierpinskiGasketGraph[n]],{n,6}])//Timing

FindSequenceFunction[seq,n]

KirchhoffSumIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"KirchhoffSumIndex"],{n,5}]

(seq=Table[KirchhoffSumIndex[SierpinskiGasketGraph[n]],{n,6}])//Timing

FindSequenceFunction[seq,n]

LaplacianMatrix

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

LaplacianPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"LaplacianPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"SierpinskiSieve",n},"LeafCount"],{n,5}]

LongestCycleCount

Same as |HC|

seq=Table[GraphData[{"SierpinskiSieve",n},"LongestCycleCount"],{n,5}]

LovaszNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"LovaszNumber"],{n,5}]

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"MatchingGeneratingPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

MatchingNumber

A122983

seq=Table[GraphData[{"SierpinskiSieve",n},"MatchingNumber"],{n,5}]

(seq=Table[MatchingNumber[SierpinskiGasketGraph[n]],{n,4}])//Timing

(seq=Table[MatchingNumber[SierpinskiGasketGraph[n]],{n,5}])//Timing

(seq=Table[MatchingNumber[SierpinskiGasketGraph[n]],{n,6}])//Timing

(seq=Table[MatchingNumber[SierpinskiGasketGraph[n]],{n,7}])//Timing

FindSequenceFunction[seq,n]//FullSimplify

MatchingPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"MatchingPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximalCliqueCount"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

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

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximalIndependentEdgeSetCount"],{n,5}]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximalIndependentVertexSetCount"],{n,5}]

MaximalIrredundantSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximalIrredundantSetCount"],{n,5}]

MaximumCliqueCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumCliqueCount"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

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

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumIndependentEdgeSetCount"],{n,5}]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumIndependentVertexSetCount"],{n,5}]

MaximumIrredundantSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumIrredundantSetCount"],{n,5}]

MaximumLeafNumber

Possibly 2^n

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumLeafNumber"],{n,5}]

MaximumVertexDegree

seq=Table[GraphData[{"SierpinskiSieve",n},"MaximumVertexDegree"],{n,5}]

(seq=Table[Max[VertexDegree[SierpinskiGasketGraph[n]]],{n,10}])//Timing

MeanDistance

seq=Table[GraphData[{"SierpinskiSieve",n},"MeanDistance"],{n,5}]

(seq=Table[Mean[Flatten@GraphDistanceMatrix[SierpinskiGasketGraph[n]]],{n,9}])//Timing

(seq=Table[Mean[Flatten@GraphDistanceMatrix[SierpinskiGasketGraph[n]]],{n,10}])//Timing

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

MinimalDominatingSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimalDominatingSetCount"],{n,5}]

MinimumCliqueCoveringCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumCliqueCoveringCount"],{n,5}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumConnectedDominatingSetCount"],{n,5}]

FindSequenceFunction[seq,n]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumDistinguishingLabelingCount"],{n,5}]

MinimumDominatingSetCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumDominatingSetCount"],{n,5}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumEdgeCoverCount"],{n,5}]

MinimumVertexDegree

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumVertexDegree"],{n,5}]

(seq=Table[Min[VertexDegree[SierpinskiGasketGraph[n]]],{n,10}])//Timing

MolecularTopologicalIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"MolecularTopologicalIndex"],{n,5}]

(seq=Table[MolecularTopologicalIndex[SierpinskiGasketGraph[n]],{n,9}])//Timing

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

Nonhamiltonian

Table[GraphData[{"SierpinskiSieve",n},"Nonhamiltonian"],{n,5}]

NormalizedLaplacianMatrix

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

OddChordlessCycleCount

seq=Table[GraphData[{"SierpinskiSieve",n},"OddChordlessCycleCount"],{n,5}]

PathCount

A288629

seq=Table[GraphData[{"SierpinskiSieve",n},"PathCount"],{n,5}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

PathPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"PathPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

PebblingNumber

seq=Table[GraphData[{"Path",n},"PebblingNumber"],{n,5}]

seq=Table[GraphData[{"SierpinskiSieve",n},"PebblingNumber"],{n,5}]

PentagonCount

seq=Table[GraphData[{"SierpinskiSieve",n},"PentagonCount"],{n,5}]

FindLinearRecurrence[seq]

FindSequenceFunction[Rest@seq,n-1]

gf=x FindGeneratingFunction[seq,x]

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

Perfect

Table[GraphData[{"SierpinskiSieve",n},"Perfect"],{n,5}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"ProjectivePlaneCrossingNumber"],{n,5}]

Radius

seq=Table[GraphData[{"SierpinskiSieve",n},"Radius"],{n,5}]

seq=Table[GraphRadius[SierpinskiGasketGraph[n]],{n,7}]

gf=x FindGeneratingFunction[seq,x]

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

RankPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"RankPolynomial"],{n,k0=1,kmax=5}]/.f_Function:>f[x,y]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"RectilinearCrossingNumber"],{n,5}]

ReliabilityPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"ReliabilityPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"SigmaPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Skewness

seq=Table[GraphData[{"SierpinskiSieve",n},"Skewness"],{n,5}]

SpanningTreeCount

A193256

seq=Table[GraphData[{"SierpinskiSieve",n},"SpanningTreeCount"],{n,5}]

Table[(3/20)^(1/4)*(5/3)^(-(n-1)/2)*(540^(1/4))^(3^(n-1)),{n,5}]

FullSimplify[(3/20)^(1/4)*(5/3)^(-(n-1)/2)*(540^(1/4))^(3^(n-1))]

seq=Table[(2^((1/6) (-3+(3^n)))) (3^((1/4) (-1+(3^n)+2 n))) (5^((1/12) (3+(3^n)-6 n))),{n,12}];

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

SquareCount

seq=Table[GraphData[{"SierpinskiSieve",n},"SquareCount"],{n,5}]

gf=x FindGeneratingFunction[seq,x]

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

StabilityIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"StabilityIndex"],{n,5}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"TopologicalIndex"],{n,5}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

seq=Table[GraphData[{"SierpinskiSieve",n},"ToroidalCrossingNumber"],{n,5}]

Traceable

Table[GraphData[{"SierpinskiSieve",n},"Traceable"],{n,5}]

TriangleCount

seq=Table[GraphData[{"SierpinskiSieve",n},"TriangleCount"],{n,5}]

gf=x FindGeneratingFunction[seq,x]

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

TuttePolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"TuttePolynomial"],{n,k0=1,kmax=5}]/.f_Function:>f[x,y]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"SierpinskiSieve",n},"Untraceable"],{n,5}]

VertexConnectivity

Table[GraphData[{"SierpinskiSieve",n},"VertexConnectivity"],{n,5}]

VertexCount

seq=Table[GraphData[{"SierpinskiSieve",n},"VertexCount"],{n,5}]

FindSequenceFunction[seq,n]//FullSimplify

VertexCoverNumber

vertex cover number == vertex count - independence number

seq=Table[GraphData[{"SierpinskiSieve",n},"VertexCoverNumber"],{n,5}]

FindSequenceFunction[Rest@seq,n-1]

VertexCoverPolynomial

(poly=Factor[Table[GraphData[{"SierpinskiSieve",n},"VertexCoverPolynomial"],{n,k0=1,kmax=5}]/.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]

Factor[Take[poly2-poly,kmax-k0-Count[poly,_Missing]+1]]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"SierpinskiSieve",n},"VertexDegrees"],{n,5}]

VertexTransitive

Table[GraphData[{"SierpinskiSieve",n},"VertexTransitive"],{n,5}]

WienerIndex

A290129

Apparently, there is a formula in the A. Hinz reference.  (Not that I can see )

A. Hinz, A. Schief, The average distance on the Sierpinski gasket, Probab. Th. Rel. Fields (1990) 87: 129.

Do[Print[{{"SierpinskiSieve",n},WienerIndex[SierpinskiGasketGraph[n]]//Timing}],{n,12}]

bug(339509): GraphDistanceMatrix[CycleGraph[250000]] returns $Aborted[]

seq=Table[GraphData[{"SierpinskiSieve",n},"WienerIndex"],{n,5}]

seq={3,21,246,3765,64032,1130463,20215254,363069729,6530385420,117517503027,2115137375634};

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

WienerSumIndex

seq=Table[GraphData[{"SierpinskiSieve",n},"WienerSumIndex"],{n,5}]

FindSequenceFunction[seq,n]

GraphData

GraphClassString

GraphClassString[GraphData["Sierpinski"]]

GraphDataString

Do[Print[GraphDataString[SierpinskiGraph[n],
"Name"->ToString[n]<>"-Sierpinski graph",
"StandardName"->{"Sierpinski",n},
"Classes"->{"Sierpinski"},
"NotationRules"->{"Sierpinski"->n},
TimeConstraint->60,
Debug->True
]],{n,6}]