Kayak Paddle Graph

Author

Eric W. Weisstein
April 25, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Figure

GraphicsGrid[Partition[Flatten[Table[Show[GraphData[{"KayakPaddle",{k,m,l}}],PlotLabel->Row[{"KP","(",k,", ",m,", ",l,")"}]],{m,3,5},{k,3,m},{l,2}]],3],BaseStyle->InsetBoxOptions->{BaseStyle->Directive[Black,FontFamily->"Times",14]},Dividers->All]

Special cases

TextGrid[{GraphData[#,"NotationRules","KayakPaddle"],#}&/@DeleteCases[GraphData["KayakPaddle"],{"KayakPaddle",_}],Dividers->All]

Properties

Acyclic

Table[GraphData[{"KayakPaddle",{k,m,l}},"Acyclic"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

AdjacencyMatrixCount

seq=Table[GraphData[{"KayakPaddle",{k,m,l}},"AdjacencyMatrixCount"],{k,3,10},{m,k,10},{l,5}]

Table[((n!)/(Piecewise[{{1,n==1},{6,n==3}},8])),{k,3,10},{m,k,10},{l,5}]

Anarboricity

Table[GraphData[{"KayakPaddle",{k,m,l}},"Anarboricity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Arboricity

Table[GraphData[{"KayakPaddle",{k,m,l}},"Arboricity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ArcTransitive

Table[GraphData[{"KayakPaddle",{k,m,l}},"ArcTransitive"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ArcTransitivity

Table[GraphData[{"KayakPaddle",{k,m,l}},"ArcTransitivity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

AutomorphismCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"AutomorphismCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

BalabanIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"BalabanIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Bandwidth

Table[GraphData[{"KayakPaddle",{k,m,l}},"Bandwidth"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

BridgeCount

Flatten[Table[GraphData[{"KayakPaddle",{k,m,l}},"BridgeCount"],{k,3,10},{m,k,10},{l,5}],1]//Union

BurningNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"BurningNumber"],{k,3,10},{m,k,10},{l,5}]

CharacteristicPolynomial

Factor@(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"CharacteristicPolynomial"],{n,k0=1,kmax=19,2}]/.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

Table[GraphData[{"KayakPaddle",{k,m,l}},"ChordCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Chordless

Table[GraphData[{"KayakPaddle",{k,m,l}},"Chordless"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ChordlessCycleCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"ChordlessCycleCount"],{k,3,10},{m,k,10},{l,5}]

Chords

Flatten[Table[GraphData[{"KayakPaddle",{k,m,l}},"Chords"],{k,3,10},{m,k,10},{l,5}],2]//Union

ChromaticInvariant

Table[GraphData[{"KayakPaddle",{k,m,l}},"ChromaticInvariant"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ChromaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"ChromaticNumber"],{k,3,10},{m,k,10},{l,5}]

ChromaticPolynomial

Factor@(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"ChromaticPolynomial"],{n,k0=1,kmax=19,2}]/.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[{"KayakPaddle",{k,m,l}},"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

DeleteCases[Flatten[Table[{Max[k,m],GraphData[{"KayakPaddle",{k,m,l}},"Circumference"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

Table[

Classes

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

GraphDataClassString["KayakPaddle"]

CliqueCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"CliqueCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]//Factor

CliqueCoveringNumber

Flatten[Table[{{k,l,m},GraphData[{"KayakPaddle",{k,m,l}},"CliqueCoveringNumber"]},{k,3,10},{m,k,10},{l,5}],2]

CliquePolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"CliquePolynomial"][x],{n,k0=1,kmax=19,2}])//Column

CliqueNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"CliqueNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ComplementOddChordlessCycleCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"ComplementOddChordlessCycleCount"],{k,3,10},{m,k,10},{l,5}]

Table[Graph[Range[n],EdgeList[GraphComplement@TriangularSnakeGraph[n]]],{n,1,9,2}]

ConnectedDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"ConnectedDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]

ConnectedDominationNumber

DeleteCases[Flatten[Table[{k+m+l-5,GraphData[{"KayakPaddle",{k,m,l}},"ConnectedDominationNumber"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

ConnectedDominationPolynomial

(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"ConnectedDominationPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Factor//Column

ConnectedInducedSubgraphCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"ConnectedInducedSubgraphCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,n]//Factor

Table[(9 2^n-6n-10)/2,{n,20}]

FindLinearRecurrence[seq]

gf=x FindGeneratingFunction[seq,x]

ConnectedInducedSubgraphPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"ConnectedInducedSubgraphPolynomial"],{n,k0=1,19,2}]/.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]

CrossingNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"CrossingNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

CycleCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"CycleCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

CyclePolynomial

DeleteCases[Flatten[Table[{x^k+x^m,GraphData[{"KayakPaddle",{k,m,l}},"CyclePolynomial"][x]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

Cyclic

Table[GraphData[{"KayakPaddle",{k,m,l}},"Cyclic"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

CyclomaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"CyclomaticNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

DetourIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"DetourIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,n]//Factor

DetourPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"DetourPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[Rest@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]

Diameter

Table[GraphData[{"KayakPaddle",{k,m,l}},"Diameter"],{k,3,10},{m,k,10},{l,5}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"DistancePolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[Rest@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

Table[GraphData[{"KayakPaddle",{k,m,l}},"DistinguishingNumber"],{k,3,10},{m,k,10},{l,5}]

DomaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"DomaticNumber"],{k,3,10},{m,k,10},{l,5}]

DominationNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"DominationNumber"],{k,3,10},{m,k,10},{l,5}]

DominationPolynomial

(poly=Factor[Table[GraphData[{"KayakPaddle",{k,m,l}},"DominationPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,5],x]

coef=Factor/@FindLinearRecurrence[Rest@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

Table[GraphData[{"KayakPaddle",{k,m,l}},"DominatingSetCount"],{k,3,10},{m,k,10},{l,5}]

LinearRecurrence[{3,2},{2,7},40]

FindLinearRecurrence[seq]

FullSimplify[FindSequenceFunction[seq,n+1],n∈Integers]

gf=FindGeneratingFunction[DeleteMissing@seq,x]

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

EdgeChromaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"EdgeChromaticNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

EdgeConnectivity

Table[GraphData[{"KayakPaddle",{k,m,l}},"EdgeConnectivity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

EdgeCoverCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"EdgeCoverCount"],{k,3,10},{m,k,10},{l,5}]

EdgeCoverNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"EdgeCoverNumber"],{k,3,10},{m,k,10},{l,5}]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"KayakPaddle",{k,m,l}},"EdgeCoverPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x]])//Column

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

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

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]

EulerianCycleCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"EulerianCycleCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FaceCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"FaceCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FlowPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"FlowPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

FractionalChromaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"FractionalChromaticNumber"],{k,3,10},{m,k,10},{l,5}]

FractionalCliqueNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"FractionalCliqueNumber"],{k,3,10},{m,k,10},{l,5}]

FractionalEdgeChromaticNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"FractionalEdgeChromaticNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Girth

DeleteCases[Flatten[Table[{Min[k,m],GraphData[{"KayakPaddle",{k,m,l}},"Girth"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

Graceful

DeleteCases[Flatten[Table[{{k,m,l},GraphData[{"KayakPaddle",{k,m,l}},"Graceful"]},{k,3,10},{m,k,10},{l,5}],2],{_,True}]

HamiltonDecomposable

Table[GraphData[{"KayakPaddle",{k,m,l}},"HamiltonDecomposable"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

HamiltonDecompositionCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"HamiltonDecompositionCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Hamiltonian

Table[GraphData[{"KayakPaddle",{k,m,l}},"Hamiltonian"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

HamiltonianCycleCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"HamiltonianCycleCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

HamiltonianNumber

Table[{{k,l,m},GraphData[{"KayakPaddle",{k,m,l}},"HamiltonianNumber"]},{k,3,10},{m,k,10},{l,5}]

HamiltonianPathCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"HamiltonianPathCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

HamiltonianWalkCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"HamiltonianWalkCount"],{k,3,10},{m,k,10},{l,5}]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

Do[Print[{n,Length@FindHamiltonianWalk[GraphData[{"KayakPaddle",{k,m,l}}],All]//Timing}],{n,1,11,2}]

HararyIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"HararyIndex"],{k,3,10},{m,k,10},{l,5}]

HexagonCount

DeleteCases[Flatten[Table[{Boole[k==6]+Boole[m==6],GraphData[{"KayakPaddle",{k,m,l}},"HexagonCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"IdiosyncraticPolynomial"],{n,k0=1,kmax=19,2}]/.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

Table[GraphData[{"KayakPaddle",{k,m,l}},"IndependenceNumber"],{k,3,10},{m,k,10},{l,5}]

IndependencePolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"IndependencePolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

CoefficientList[Take[poly,4],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]]

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

IndependenceRatio

Table[GraphData[{"KayakPaddle",{k,m,l}},"IndependenceRatio"],{k,3,10},{m,k,10},{l,5}]

IndependentEdgeSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"IndependentEdgeSetCount"],{k,3,10},{m,k,10},{l,5}]

IndependentVertexSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"IndependentVertexSetCount"],{k,3,10},{m,k,10},{l,5}]

FullSimplify[FindSequenceFunction[seq,(n+1)/2],n∈Integers]

IntersectionNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"IntersectionNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

IrredundancePolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"IrredundancePolynomial"],{n,k0=1,19,2}]/.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}]

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

Table[GraphData[{"KayakPaddle",{k,m,l}},"IrredundantSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[DeleteMissing@seq,n]

KirchhoffIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"KirchhoffIndex"],{k,3,10},{m,k,10},{l,5}]

KirchhoffSumIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"KirchhoffSumIndex"],{k,3,10},{m,k,10},{l,5}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"LaplacianPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

LeafCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"LeafCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Likelihood

Table[GraphData[{"KayakPaddle",{k,m,l}},"Likelihood"],{k,3,10},{m,k,10},{l,5}]

LongestCycleCount

DeleteCases[Flatten[Table[{1+Boole[k==m],GraphData[{"KayakPaddle",{k,m,l}},"LongestCycleCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

LongestPathLength

Table[GraphData[{"KayakPaddle",{k,m,l}},"LongestPathLength"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

LongestPathCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"LongestPathCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

LovaszNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"LovaszNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MatchingGeneratingPolynomial

Factor@(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"MatchingGeneratingPolynomial"],{n,k0=1,kmax=19,2}]/.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>2]

MatchingNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"MatchingNumber"],{k,3,10},{m,k,10},{l,5}]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"MatchingPolynomial"],{n,k0=1,kmax=19,2}]/.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}]//FullSimplify

MaximalCliqueCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximalCliqueCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximalIndependencePolynomial

(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximalIndependencePolynomial"],{n,k0=1,19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[Rest@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}]//FullSimplify

MaximalIndependentEdgeSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximalIndependentEdgeSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]//FullSimplify

InputForm[%]

gf=x FindGeneratingFunction[seq,x]

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

MaximalIndependentVertexSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximalIndependentVertexSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[Rest@seq,(n-1)/2]//FullSimplify

Table[Piecewise[{{1,n==1}},2 Fibonacci[(1/2) (-1+n)]+LucasL[(1/2) (-1+n)]],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximalIrredundantSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximalIrredundantSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Table[RootSum[-1-#-#^2+#^3&,2 #^n-4 #^(n+1)+3 #^(n+2)&]/11,{n,2,20}]

MaximumCliqueCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumCliqueCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximumIndependentEdgeSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumIndependentEdgeSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximumIndependentVertexSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumIndependentVertexSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximumIrredundantSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumIrredundantSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MaximumLeafNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumLeafNumber"],{k,3,10},{m,k,10},{l,5}]

MaximumVertexDegree

Table[GraphData[{"KayakPaddle",{k,m,l}},"MaximumVertexDegree"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MeanDistance

Table[GraphData[{"KayakPaddle",{k,m,l}},"MeanDistance"],{k,3,10},{m,k,10},{l,5}]

FindSequenceFunction[seq,(n+1)/2]//Factor

MinimalDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimalDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MinimalDominationPolynomial

(poly=Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimalDominationPolynomial"],{n,k0=1,19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[Rest@poly]

MinimalEdgeCoverCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimalEdgeCoverCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[DeleteMissing@seq,n]

InputForm[%]

f1=-1-#1-2*#1^2+#1^3&;
f2=-25+82*#1-87*#1^2+87*#1^3&;
pos={1,2,3};

(an=ToNumberField[Root[f2,#],Root[f1,pos[[#]]]]&/@Range[3])//InputForm

coeffs=an[[All,-1]]

SameQ@@coeffs

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

Table[-RootSum[-1-#1-2 ((#1)^2)+((#1)^3)&,-43 ((#1)^n)-99 ((#1)^(n+1))+40 ((#1)^(n+2))&]/87,{n,50}]

coef=FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{2,1,1},{0,3,7},20]

FindSequenceFunction[seq,n]

gf=x Factor@FindGeneratingFunction[seq,x]

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

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

InputForm[%]

MinimalTotalDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimalTotalDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

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

coef=FindLinearRecurrence[DeleteMissing@seq]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

MinimumCoveringsByMaximalCliquesCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumCoveringsByMaximalCliquesCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MinimumConnectedDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumConnectedDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MinimumDistinguishingLabelingCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumDistinguishingLabelingCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteCases[seq,_Missing],x]

MinimumDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MinimumEdgeCoverCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumEdgeCoverCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[DeleteMissing@seq,(n+1)/2]//Factor

MinimumTotalDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumTotalDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

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

MinimumVertexCoverCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumVertexCoverCount"],{k,3,10},{m,k,10},{l,5}]

MinimumVertexDegree

Table[GraphData[{"KayakPaddle",{k,m,l}},"MinimumVertexDegree"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

MolecularTopologicalIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"MolecularTopologicalIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindLinearRecurrence[seq]

FindSequenceFunction[Rest@seq,n]//Factor

Nonhamiltonian

Table[GraphData[{"KayakPaddle",{k,m,l}},"Nonhamiltonian"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

OddChordlessCycleCount

DeleteCases[Flatten[Table[{Total[Boole[OddQ[DeleteCases[{k,m},3]]]],GraphData[{"KayakPaddle",{k,m,l}},"OddChordlessCycleCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

PathCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"PathCount"],{k,3,10},{m,k,10},{l,5}]

PathPolynomial

(poly=Factor[Table[GraphData[{"KayakPaddle",{k,m,l}},"PathPolynomial"],{n,k0=1,kmax=19,2}]/.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]

PentagonCount

DeleteCases[Flatten[Table[{Boole[k==5]+Boole[m==5],GraphData[{"KayakPaddle",{k,m,l}},"PentagonCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

Perfect

Table[GraphData[{"KayakPaddle",{k,m,l}},"Perfect"],{k,3,10},{m,k,10},{l,5}]

Planar

Table[GraphData[{"KayakPaddle",{k,m,l}},"Planar"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ProjectivePlaneCrossingNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"ProjectivePlaneCrossingNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Radius

Table[{{k,m,l},GraphData[{"KayakPaddle",{k,m,l}},"Radius"]},{k,3,10},{m,k,10},{l,5}]

RankPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"RankPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x,y])//Column

RectilinearCrossingNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"RectilinearCrossingNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"ReliabilityPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

ShannonCapacity

Table[GraphData[{"KayakPaddle",{k,m,l}},"ShannonCapacity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

SigmaPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"SigmaPolynomial"],{n,k0=1,kmax=19,2}]/.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

Table[GraphData[{"KayakPaddle",{k,m,l}},"Skewness"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

SpanningTreeCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"SpanningTreeCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]//FullSimplify

SquareCount

DeleteCases[Flatten[Table[{Boole[k==4]+Boole[m==4],GraphData[{"KayakPaddle",{k,m,l}},"SquareCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

StabilityIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"StabilityIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]

FindGeneratingFunction[seq,x]

TopologicalIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"TopologicalIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"ToroidalCrossingNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

TotalDominatingSetCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"TotalDominatingSetCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindLinearRecurrence[Rest@seq]

LinearRecurrence[{2,3,2},{1,4,16},40]

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

TotalDominationNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"TotalDominationNumber"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

TotalDominationPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"TotalDominationPolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[Rest@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[{"KayakPaddle",{k,m,l}},"Traceable"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

Triameter

Table[GraphData[{"KayakPaddle",{k,m,l}},"Triameter"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

TriangleCount

DeleteCases[Flatten[Table[{Boole[k==3]+Boole[m==3],GraphData[{"KayakPaddle",{k,m,l}},"TriangleCount"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

TuttePolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"TuttePolynomial"],{n,k0=1,kmax=19,2}]/.f_Function:>f[x,y])//Column

Untraceable

Table[GraphData[{"KayakPaddle",{k,m,l}},"Untraceable"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

VertexConnectivity

Table[GraphData[{"KayakPaddle",{k,m,l}},"VertexConnectivity"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

VertexCoverCount

Table[GraphData[{"KayakPaddle",{k,m,l}},"VertexCoverCount"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FullSimplify[FindSequenceFunction[seq,(n+1)/2],n∈Integers]

VertexCoverNumber

Table[GraphData[{"KayakPaddle",{k,m,l}},"VertexCoverNumber"],{k,3,10},{m,k,10},{l,5}]

VertexCount

DeleteCases[Flatten[Table[{k+m+l-1,GraphData[{"KayakPaddle",{k,m,l}},"V"]},{k,3,10},{m,k,10},{l,5}],2],{x_,x_}]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"KayakPaddle",{k,m,l}},"VertexCoverPolynomial"],{n,k0=1,kmax=19,2}]/.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}]

VertexDegrees

(Table[GraphData[{"KayakPaddle",{k,m,l}},"VertexDegrees"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union)//Column

WienerIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"WienerIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]//Factor

WienerSumIndex

Table[GraphData[{"KayakPaddle",{k,m,l}},"WienerSumIndex"],{k,3,10},{m,k,10},{l,5}]//Flatten//Union

FindSequenceFunction[seq,(n+1)/2]//Factor

GraphData

GraphDataClassString

GraphDataClassString["KayakPaddle"]

GraphDataString

file=OpenAppend["~/kayak-paddle.m"]

todo=Complement[Flatten[Table[{k,m,l},{m,3,10},{k,3,m},{l,5}]//Flatten//Union,2],rules[[All,-1]]]

Length[todo]

WriteString[file,GraphDataString[KayakPaddleGraph@@#,
"StandardName"->{"KayakPaddle",#},
"Name"->StringReplace[ToString[#,InputForm],{" "->"","{"->"(","}"->")"}]<>"\[Hyphen]kayak paddle graph",
"Information"->"KayakPaddleGraph",
"Classes"->{"KayakPaddle"},
"NotationRules"->{"KayakPaddle"->#},
"Notation"->Row[{"KP","(",#[[1]],",",#[[2]],",",#[[2]],")"}],
"Skip"->{"ComplementOddChordlessCycle","DomaticNumber","MinimumWeightFractionalColoring"}
]]&/@Take[todo,All];

Close[file]