Gear Graph

Author

Eric W. Weisstein
May 1, 2018

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

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

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

Graphs

GraphData default

GraphicsRow[Table[GraphData[{"Gear",n},"Graph","Gear"][[1]],{n,3,8}],Dividers->All,ImageSize->500]//StyleGraphs[#,VertexSize->{"Scaled",.04}]&

GraphData "UnitDistance"

GraphicsRow[Table[GraphData[{"Gear",n},"Graph","UnitDistance"][[1]],{n,3,8}],Dividers->All,ImageSize->500]//StyleGraphs[#,VertexSize->{"Scaled",.04}]&

Graphs.m

GraphicsGrid[Partition[Table[Show[GearGraph[n,VertexSize->{"Scaled",.04},System`VertexStyle->Red,System`EdgeStyle->Black],PlotLabel->Style[HoldForm[n]==n,FontFamily->"Times"]],{n,3,8}],UpTo[3]],Dividers->All,ImageSize->400]

Special cases

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

Extra unit-distance graphs

UnitDistanceGearGraph[n_]:=Module[{v=GraphEmbedding[GearGraph[n,GraphLayout->"UnitDistance"]],e,nn=2n+1},
e=Quiet[Select[Subsets[Range[nn],{2}],SquaredEuclideanDistance@@v[[#]]==1.&],N::meprec];
System`Graph[Range[nn],UndirectedEdge@@@e,VertexCoordinates->v]
]

new=DeleteCases[Table[n->If[IsomorphicGraphQ[GearGraph[n],ud=UnitDistanceGearGraph[n]],{},ud],{n,3,20}],HoldPattern[_->{}]]

(#->RecognizeGraph[#2])&@@@new

DegenerateEmbeddingQ/@new[[All,-1]]

VertexCount/@new[[All,-1]]

GraphicsRow[Show[StyleGraphs[#2],PlotLabel->Style[(n==#1),Directive[Black,FontFamily->"Times"]]]&@@@new,ImageSize->500]

GraphData[{"GearUnitDistance",#}]&/@{3,6,12,18}

Properties

Acyclic

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

AdjacencyMatrixCount

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

FindSequenceFunction[seq,n-2]//FullSimplify

Arboricity

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

ArcTransitive

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

ArcTransitivity

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

AutomorphismCount

Table[GraphData[{"Gear",n},"AutomorphismCount"],{n,3,20}]

BalabanIndex

Table[GraphData[{"Gear",n},"BalabanIndex"],{n,3,20}]

Bandwidth

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

BridgeCount

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

BurningNumber

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

CharacteristicPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

ChordlessCycleCount

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

FindSequenceFunction[seq,n-2]

ChromaticInvariant

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

ChromaticNumber

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

ChromaticPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Circumference

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

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n-2]//Factor

CliqueCoveringNumber

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

CliqueNumber

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

CliquePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

A287431

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

FindLinearRecurrence[seq]

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

LinearRecurrence[{4,-1,-6,4},{7,21,45,129},20]

Table[(2^n)LucasL[n]+4n+1,{n,20}]

ConnectedDominationNumber

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

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

Table[1+Ceiling[n/2],{n,3,20}]

ConnectedDominationPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing[poly]]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

Table[(x^(n/2))(((1+x))^n)LucasL[n,(Sqrt[x])],{n,5}]//Factor//Column

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

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

(poly2=Table[(x^(2 n))+2 n (x^(-2+2 n)) (1+x)+(x^(n/2+1))(((1+x))^n)LucasL[n,(Sqrt[x])],{n,3,11}]//Factor)//Column

FullSimplify[poly2-DeleteMissing[poly]]

FullSimplify[(x^(2 n))+2 n (x^(-2+2 n)) (1+x)+(x^(n/2+1))(((1+x))^n)LucasL[n,(Sqrt[x])]]

ConnectedInducedSubgraphCount

A286188(n) == A206776(n)+4(n^2) - 2n + 1

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

FindLinearRecurrence[DeleteMissing[seq]]

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

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

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

Table[((((1/2) (3-(Sqrt[17]))))^n)+((((1/2) (3+(Sqrt[17]))))^n),{n,10}]//FullSimplify

ConnectedInducedSubgraphPolynomial

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

Length[DeleteMissing[poly]]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

coef=Factor/@FindLinearRecurrence[Drop[DeleteMissing@poly],2]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Take[poly,4]//Expand//Column

Table[(4 (n^2)-2n)(x^n),{n,3,6}]//FullSimplify//Column

poly2=Table[(x^n)(((((1/2) (3-(Sqrt[17]))))^n)+((((1/2) (3+(Sqrt[17]))))^n)),{n,3,7}]//FullSimplify//Column

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]//Simplify

CyclePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

Cyclic

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

CyclomaticNumber

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

DetourIndex

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

FindSequenceFunction[seq,n-2]//FullSimplify

DetourMatrix

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

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

DetourPolynomial

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

FindSequenceFunction[{156,404,830,1482,2408},n-2]//Simplify

FindSequenceFunction[{26,50,82,122,170},n-2]//Factor

coef=Factor/@FindLinearRecurrence[poly]

poly2=Table[-(((2n-2+x))^(n-1))(((2n+x))^n)(-n (1-7 n+8 (n^2))-2 (1-2 n+2 (n^2))x+(x^2)),{n,k0,kmax}]

FullSimplify[poly2-poly]

Diameter

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

DistanceMatrix

MatrixForm/@Table[GraphData[{"Gear",n},"DistanceMatrix"],{n,3,5}]

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

DistancePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

DistinguishingNumber

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

DomaticNumber

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

DominatingSetCount

A290938

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

coef=FindLinearRecurrence[seq]

seq=LinearRecurrence[{7,-12,-2,3,3,2},{5,23,83,291,1015,3539},40]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((-5+12 x+18 (x^2)+4 (x^3)-5 (x^4)-8 (x^5))/(-1+7 x-12 (x^2)-2 (x^3)+3 (x^4)+3 (x^5)+2 (x^6))),{x,0,20}],x]

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

Table[((((1/2) (3-(Sqrt[17]))))^n)+((((1/2) (3+(Sqrt[17]))))^n)-1+RootSum[-1-#-3 ((#)^2)+((#)^3)&,#^n&],{n,20}]//Expand

DominationNumber

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

DominationPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EdgeChromaticNumber

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

EdgeConnectivity

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

EdgeCount

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

FindSequenceFunction[seq,n-2]

EdgeCoverCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]//FullSimplify

EdgeCoverNumber

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

EdgeCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EulerianCycleCount

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

FaceCount

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

FlowPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

Probably 2(n+1)

IndependenceNumber

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

IndependencePolynomial

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

CoefficientList[Take[poly,5],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]]

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

Take[poly,5]//Column

(poly2=Expand@Table[x(((x+1))^n)+(x^n)LucasL[2n,1/(Sqrt[x])],{n,3,5}])//Column

Take[poly,5]//Column

poly2=Table[x(((1+x))^n)+(x^n)LucasL[2n,1/(Sqrt[x])],{n,k0,kmax}];

FullSimplify[poly2-poly]

IndependentEdgeSetCount

A287349

(x^n) (n (Sqrt[x]) Fibonacci[2 n,(1/(Sqrt[x]))]+LucasL[2 n,(1/(Sqrt[x]))])/.x->1

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

FindLinearRecurrence[DeleteMissing[seq]]

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

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

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

Table[(1/(2^n))((((3-(Sqrt[5])))^n) +(((3+(Sqrt[5])))^n) )+(n/((2^n)(Sqrt[5])))((((3+(Sqrt[5])))^n)-(((3-(Sqrt[5])))^n)),{n,10}]//FullSimplify

Table[(1/2)((((3-(Sqrt[5])))^n) +(((3+(Sqrt[5])))^n) ),{n,10}]//FullSimplify

Table[2^(n-1)LucasL[2n],{n,10}]

Table[(1/(2(Sqrt[5])))(-(((3-(Sqrt[5])))^n) +(((3+(Sqrt[5])))^n) ),{n,10}]//FullSimplify

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

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

LinearRecurrence[{6,-11,6,-1},{4,13,42,131},30]

seq

IndependentVertexSetCount

A287350

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

FindLinearRecurrence[DeleteMissing[seq]]

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

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

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

Table[(2^n)+((((1/2) (3-(Sqrt[5]))))^n)+((((1/2) (3+(Sqrt[5]))))^n),{n,20}]//FullSimplify

Table[((((1/2) (3-(Sqrt[5]))))^n)+((((1/2) (3+(Sqrt[5]))))^n),{n,10}]//Expand

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

Table[(2^n)+LucasL[2n],{n,30}]

LinearRecurrence[{5,-7,2},{5,11,26},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

IrredundantSetCount

A290589

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

seq={41,113,320,908,2574,7317,20855,59582,170681,490180,1411207,4072092,11774998,34114389,99006951,287783255};

Table[FindLinearRecurrence[Drop[DeleteMissing@seq,k]],{k,0,3}]

Table[FindSequenceFunction[Drop[DeleteMissing@seq,k],n-2-k],{k,0,3}]

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

MatchingGeneratingPolynomial

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

CoefficientList[Take[poly,5],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]]

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

FullSimplify[FunctionExpand[LucasL[2n,(Sqrt[x])]],n∈Integers]

(poly2=Expand@Table[((x^n)LucasL[2n,(Sqrt[x])]+n (x^(n-1)) (Sqrt[x]) Fibonacci[2n,(Sqrt[x])]),{n,3,5}])//Column

(poly2=Expand@Table[(x^(2n))((x^(-n))LucasL[2n,1/(Sqrt[x])]+n (x^(-(n-1))) 1/(Sqrt[x]) Fibonacci[2n,1/(Sqrt[x])]),{n,3,5}])//Column

FullSimplify[(x^(2n))((x^(-n))LucasL[2n,1/(Sqrt[x])]+n (x^(-(n-1))) 1/(Sqrt[x]) Fibonacci[2n,1/(Sqrt[x])])]

Take[poly,5]//Column

poly2=Table[(x^n) (n (Sqrt[x]) Fibonacci[2 n,(1/(Sqrt[x]))]+LucasL[2 n,(1/(Sqrt[x]))]),{n,k0,kmax}];

FullSimplify[poly2-poly]

MatchingNumber

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

MatchingPolynomial

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

Take[poly,5]//Column

FullSimplify[(x^n)((x^n) (n (Sqrt[x]) Fibonacci[2 n,(1/(Sqrt[x]))]+LucasL[2 n,(1/(Sqrt[x]))])/.x->-1/x^2),x∈Reals&&n ∈Integers&&n>0]

Table[(x^(n+1))(((-(1/x)))^n) (((I n Fibonacci[2 n,-I Abs[x]])/(Abs[x]))+LucasL[2 n,-I Abs[x]]),{n,3,5}]//FullSimplify[#,x>0]&//Column

Table[(((-1))^n)(I n Fibonacci[2 n,-I x]+x LucasL[2 n,-I x]),{n,3,5}]//Factor//Column

MaximalCliqueCount

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

FindSequenceFunction[seq,n-2]

MaximalIndependentEdgeSetCount

A287425

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

FindLinearRecurrence[seq]

LinearRecurrence[{5,-10,12,-11,7,-3,1},{3,6,11,22,47,98,198},30]

FindSequenceFunction[seq,n-2]

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

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

Table[2+(1/12167)n (Root[-78310985281+57927087 #1-12167 ((#1)^2)+((#1)^3)&,1] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(2+n))+Root[-78310985281+6436343 #1+((#1)^3)&,2] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n))+Root[-78310985281+6436343 #1+((#1)^3)&,3] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n))),{n,1,3}]//FullSimplify

seq={3,6,11,22,47,98,198,394,776,1512,2917,5582,10610,20050,37697,70562,131565,244460,452829,836482};

gf=x FindGeneratingFunction[seq,x]

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

FullSimplify[{Root[-78310985281+57927087 #1-12167 ((#1)^2)+((#1)^3)&,1] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^2),Root[-78310985281+6436343 #1+((#1)^3)&,2] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^1),Root[-78310985281+6436343 #1+((#1)^3)&,3] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^1)}]

Table[2+(1/12167)n (Root[-78310985281-32181715 #1+((#1)^3)&,1]((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^n)+Root[-78310985281-32181715 #1+((#1)^3)&,3] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^n)+Root[-78310985281-32181715 #1+((#1)^3)&,2]((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^n)),{n,10}]//FullSimplify

FullSimplify[2+(1/12167)n (Root[-78310985281-32181715 #1+((#1)^3)&,1]((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^n)+Root[-78310985281-32181715 #1+((#1)^3)&,3] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^n)+Root[-78310985281-32181715 #1+((#1)^3)&,2]((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^n))]

Table[2+n (((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^n)-((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^n)) Root[1-5 #1+23 ((#1)^3)&,2]+n (((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^n)-((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^n)) Root[1-5 #1+23 ((#1)^3)&,3],{n,10}]//FullSimplify

MaximalIndependentVertexSetCount

A259967

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

FindLinearRecurrence[seq]

FullSimplify[FindSequenceFunction[seq,n-2]]

FullSimplify[Root[-1-3 #1-2 ((#1)^2)+((#1)^3)&,1] ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(-2))]

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

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

MaximumCliqueCount

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

FindSequenceFunction[seq,n-2]

MaximumIndependentEdgeSetCount

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

FindSequenceFunction[seq,n-2]//Simplify

MaximumIndependentVertexSetCount

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

MaximumLeafNumber

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

FindSequenceFunction[seq,n-2]//FullSimplify

MaximumVertexDegree

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

MinimalDominatingSetCount

A290378

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

seq={2,8,8,16,37,80,156,304,602,1173,2290,4456,8686,16892,32833,63776,123864,240524,467060,907061,1761894,3423164,6652706,12933280,25151787,48931280,95228360,185400336,361093444,703546005,1371282460,2673742784,5215147858};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{4,-3,-4,4,-1,1,3,-3,0,2,3,0,-1},{2,8,8,16,37,80,156,304,602,1173,2290,4456,8686},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

gf=((x (2-18 (x^2)+16 (x^3)+21 (x^4)-18 (x^5)-15 (x^6)-2 (x^7)+16 (x^8)+2 (x^9)+11 (x^10)-2 (x^11)-5 (x^12)))/((1+(x^2)) (1-x-(x^2)) (1-x-(x^2)-(x^3)) (1-2 x-(x^2)+3 (x^3)-(x^4)-2 (x^5)+(x^6))));

CoefficientList[Series[((2-18 (x^2)+16 (x^3)+21 (x^4)-18 (x^5)-15 (x^6)-2 (x^7)+16 (x^8)+2 (x^9)+11 (x^10)-2 (x^11)-5 (x^12))/((1+(x^2)) (1-x-(x^2)) (1-x-(x^2)-(x^3)) (1-2 x-(x^2)+3 (x^3)-(x^4)-2 (x^5)+(x^6)))),{x,0,20}],x]

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

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

FullSimplify[RootReduce[-(((-I))^n)-(I^n)-((((1/2) (1-(Sqrt[5]))))^n)-((((1/2) (1+(Sqrt[5]))))^n)+(((Root[-1-#1-((#1)^2)+((#1)^3)&,2])^n) (1+2 Root[-1-#1-((#1)^2)+((#1)^3)&,2]-3 ((Root[-1-#1-((#1)^2)+((#1)^3)&,2])^2)) (Root[-1-#1-((#1)^2)+((#1)^3)&,1]-Root[-1-#1-((#1)^2)+((#1)^3)&,3])+((Root[-1-#1-((#1)^2)+((#1)^3)&,1])^n) (-1-2 Root[-1-#1-((#1)^2)+((#1)^3)&,1]+3 ((Root[-1-#1-((#1)^2)+((#1)^3)&,1])^2)) (Root[-1-#1-((#1)^2)+((#1)^3)&,2]-Root[-1-#1-((#1)^2)+((#1)^3)&,3])+(Root[-1-#1-((#1)^2)+((#1)^3)&,1]-Root[-1-#1-((#1)^2)+((#1)^3)&,2]) ((Root[-1-#1-((#1)^2)+((#1)^3)&,3])^n) (-1-2 Root[-1-#1-((#1)^2)+((#1)^3)&,3]+3 ((Root[-1-#1-((#1)^2)+((#1)^3)&,3])^2)))/((Root[-1-#1-((#1)^2)+((#1)^3)&,1]-Root[-1-#1-((#1)^2)+((#1)^3)&,2]) (Root[-1-#1-((#1)^2)+((#1)^3)&,1]-Root[-1-#1-((#1)^2)+((#1)^3)&,3]) (Root[-1-#1-((#1)^2)+((#1)^3)&,2]-Root[-1-#1-((#1)^2)+((#1)^3)&,3]))]]

terms=List@@((((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1])^5)))/((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]))+(((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2])^5)))/((-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]))+(((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3])^5)))/((-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]))+(((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4])^5)))/((-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]))+(((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5])^5)))/((-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]) (Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]))+(((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])^n) (-2-2 Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]+9 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])^2)-4 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])^3)-10 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])^4)+6 ((Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])^5)))/((-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,1]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,2]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,3]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,4]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6]) (-Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,5]+Root[1-2 #1-((#1)^2)+3 ((#1)^3)-((#1)^4)-2 ((#1)^5)+((#1)^6)&,6])));

N[terms/.n->0,20]//Chop

terms[[All,1]]

Table[RootSum[-1-#-((#)^2)+((#)^3)&,#^n&]+RootSum[1-2 #-((#)^2)+3 ((#)^3)-((#)^4)-2 ((#)^5)+((#)^6)&,#^n&]-LucasL[n]-2 Cos[((n π)/2)],{n,20}]

MinimalDominatingSetSignature

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

FindSequenceFunction[DeleteMissing@seq,n-2]

MinimalEdgeCoverCount

A290762

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

seq={2,6,15,38,90,200,434,934,1995,4237,8976,19010,40300,85591,182231,389094,833306,1790141,3857190,8334719,18057605,39217422,85357692,186142694,406619110,889555565,1948564239,4273011841,9379101468,20603197661,45289832230,99612356518};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{6,-14,19,-21,18,-11,7,-2,1},{2,6,15,38,90,200,434,934,1995},20]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

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

FullSimplify[RootReduce[res]]

Expand[%81]//RootReduce

Root[-1+2 #1+((#1)^3)&,1] ((Root[-1-2 ((#1)^2)+((#1)^3)&,1])^(1+n))+Root[-1+2 #1+((#1)^3)&,3] ((Root[-1-2 ((#1)^2)+((#1)^3)&,2])^(1+n))+Root[-1+2 #1+((#1)^3)&,2] ((Root[-1-2 ((#1)^2)+((#1)^3)&,3])^(1+n))+

((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,1]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,2]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,3]+

n(((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,2]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,3]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(2+n)) Root[-5+18 #1+23 ((#1)^2)+23 ((#1)^3)&,1])

Root[-1+2 #1+((#1)^3)&,1] ((Root[-1-2 ((#1)^2)+((#1)^3)&,1])^(1+n))+Root[-1+2 #1+((#1)^3)&,3] ((Root[-1-2 ((#1)^2)+((#1)^3)&,2])^(1+n))+Root[-1+2 #1+((#1)^3)&,2] ((Root[-1-2 ((#1)^2)+((#1)^3)&,3])^(1+n))+

f1=-1-2 ((#1)^2)+((#1)^3)&;
f2=-1+2 #1+((#1)^3)&;
pos={1,3,2};

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@an[[All,-1]]

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]])).an[[1,2]]]&]

Table[Root[-1+2 #1+((#1)^3)&,1] ((Root[-1-2 ((#1)^2)+((#1)^3)&,1])^(1+n))+Root[-1+2 #1+((#1)^3)&,3] ((Root[-1-2 ((#1)^2)+((#1)^3)&,2])^(1+n))+Root[-1+2 #1+((#1)^3)&,2] ((Root[-1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)),{n,10}]//N//Chop

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

((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,1]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,2]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,3]

f1=-1+#1-2 ((#1)^2)+((#1)^3)&;
f2=1+2 #1+((#1)^2)+((#1)^3)&;
pos={1,2,3};

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@an[[All,-1]]

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]])).an[[1,2]]]&]

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

Table[((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,1]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,2]+((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[1+2 #1+((#1)^2)+((#1)^3)&,3],{n,10}]//N//Chop

n(((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,2]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,3]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(2+n)) Root[-5+18 #1+23 ((#1)^2)+23 ((#1)^3)&,1])

f1=-1+#1-2 ((#1)^2)+((#1)^3)&;
f2=-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&;
pos={1,3,2};

an=ToNumberField[Root[f2,pos[[#]]],Root[f1,#]]&/@Range[Length[pos]]

Equal@@an[[All,-1]]

RootSum[f1,Evaluate[((#)^(n+Range[Length[pos]])).an[[1,2]]]&]

Table[n(((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,3])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,2]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,2])^(1+n)) Root[-5+2 #1+23 ((#1)^2)+23 ((#1)^3)&,3]+ ((Root[-1+#1-2 ((#1)^2)+((#1)^3)&,1])^(2+n)) Root[-5+18 #1+23 ((#1)^2)+23 ((#1)^3)&,1]),{n,20}]//N//Chop

Table[(n/23) RootSum[-1+#1-2 ((#1)^2)+((#1)^3)&,-7 ((#1)^(1+n))-14 ((#1)^(2+n))+13 ((#1)^(3+n))&],{n,20}]

Table[RootSum[-1-2 ((#1)^2)+((#1)^3)&,-2 ((#1)^(n+2))+((#1)^(n+3))&]-RootSum[-1+#1-2 ((#1)^2)+((#1)^3)&,((#1)^(n+1))-2 ((#1)^(n+2))+((#1)^(n+3))&]+(1/23) n RootSum[-1+#1-2 ((#1)^2)+((#1)^3)&,-7 ((#1)^(n+1))-14 ((#1)^(n+2))+13 ((#1)^(n+3))&],{n,20}]

MinimumConnectedDominatingSetCount

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

Table[Piecewise[{{2,Mod[n,2]==0},{n,Mod[n,2]==1}}],{n,3,20}]

MinimumVertexDegree

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

MeanDistance

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

FindSequenceFunction[seq,n-2]//Simplify

MinimumCliqueCoveringCount

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

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

MinimumConnectedDominatingSetCount

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

FindSequenceFunction[seq,n-2]//FullSimplify

FullSimplify[%,Mod[n,2]==#]&/@{0,1}

Table[Piecewise[{{2,Mod[n,2]==0},{n,Mod[n,2]==1}}],{n,3,20}]

MinimumDistinguishingLabelingCount

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

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n-2]

MinimumDominatingSetCount

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

seq[[1;;All;;2]]

FindSequenceFunction[seq,n-2]

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

Table[Piecewise[{{3,n==3},{10,n==4},{35,n==5},{5,n==6},{28,n==7},{21,n==9},{2,Mod[n,2]==0},{2n,Mod[n,2]==1}}],{n,3,20}]

%==seq

MinimumEdgeCoverCount

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

FindSequenceFunction[seq,n-2]//Simplify

MinimumVertexDegree

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

MolecularTopologicalIndex

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

FindSequenceFunction[seq,n-2]//Simplify

Nonhamiltonian

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

NormalizedLaplacianMatrix

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

OddChordlessCycleCount

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

PathCount

A292000

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

coef=FindLinearRecurrence[seq]

seq=LinearRecurrence[{5,-10,10,-5,1},{117,322,745,1515,2793},{-1,1}]

seq=LinearRecurrence[{5,-10,10,-5,1},{5,33,117,322,745},40]

FindSequenceFunction[seq,n]//Factor

Table[n (-21+58 n-15 (n^2)+8 (n^3))/6,{n,20}]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((-5-8 x-2 (x^2)-17 (x^3))/(((-1+x))^5)),{x,0,20}],x]

PathPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

PentagonCount

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

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

RectilinearCrossingNumber

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

ReliabilityPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SigmaPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SpanningTreeCount

A003500(n) - 2

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

FindSequenceFunction[seq,n-2]//FullSimplify

Table[(((2-(Sqrt[3])))^n)+(((2+(Sqrt[3])))^n),{n,10}]//FullSimplify

Table[LucasL[2n,Sqrt[2]]-2,{n,20}]//FunctionExpand

Table[LucasL[2n,Sqrt[2]]-2,{n,20}]//Round

SquareCount

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

StabilityIndex

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

TopologicalIndex

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

FindSequenceFunction[seq[[1;;All;;2]],n]//FullSimplify

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

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

TuttePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n-2]//Simplify

VertexCoverNumber

Related to but not the same as IndependenceNumber

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

VertexCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

VertexDegrees

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

VertexTransitive

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

WienerIndex

A049598

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

FindSequenceFunction[seq,n-2]//Factor

Table[6n(n+1),{n,0,20}]

12PolygonalNumber[3,Range[0,20]]

12Binomial[Range[20],2]

FindLinearRecurrence[seq]

LinearRecurrence[{3,-3,1},{12,36,72},{0,20}]

WienerSumIndex

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

FindSequenceFunction[seq,n-2]

GraphData

GraphClassString

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

GraphDataString

Gear graphs

Table[{n,RecognizeGraph[GearGraph[n]]},{n,3,6}]

Do[Print[GraphDataString[
GearGraph[n],
"StandardName"->{"Gear",n},
"Name"->ToString[n]<>"-gear graph",
"NotationRules"->{"Gear"->n},
"Classes"->{"Gear"},
TimeConstraint->60
]],{n,3,20}]

Derived unit-distance graphs

file=OpenWrite["~/gearud.m"]

WriteString[file,GraphDataString[new[[2,2]],
"StandardName"->"StarPolyiamondGraph",
"Name"->"star polyiamond graph",
"Classes"->{"Matchstick","UnitDistance"},
"Information"->"Polyiamond",
Debug->True,
TimeConstraint->60]]

WriteString[file,GraphDataString[#[[2]],
"StandardName"->{"GearUnitDistance",#[[1]]},
"Name"->ToString[#[[1]]]<>"\[Hyphen]gear unit\[Hyphen]distance graph",
Debug->True,
TimeConstraint->60]]&/@Drop[new,2]

Close[file]