Wheel Complement Graph

Author

Eric W. Weisstein
November 30, 2022

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

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

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

Special cases

<<MathWorld`Graphs`

TextGrid[Sort[{GraphData[#,"NotationRules","WheelComplement"],#}&/@GraphData["WheelComplement"]],Dividers->All]

GraphicsRow[Table[Show[StyleGraphs@GraphData[{"WheelComplement",n}],PlotLabel->Style[Subscript[OverBar[Style["W",Italic]],n],Directive[14,Black,FontFamily->"Times"]]],{n,4,8}],Dividers->All]

WheelComplementGraph[n_Integer]:=Module[{g=CirculantGraph1[n-1,Range[2,Floor[n/2]]],v},
Graph[Range[n],EdgeList[g],VertexCoordinates->Append[v=GraphEmbedding[g],{0,-1}/5+Min/@Transpose[v]]]
]/;n >4

Table[IsomorphicGraphQ[GraphComplement[WheelGraph[n]],WheelComplementGraph[n]],{n,4,20}]

Table[WheelComplementGraph[n],{n,5,10}]

data=Table[{
n,
g=GraphComplement[WheelGraph[n]],
CanonicalGraph@g,
RecognizeGraph@g,
RecognizeGraph/@ConnectedGraphComponents[g],
Length@ConnectedComponents[g]},{n,4,22}];

TextGrid[data,Dividers->All]

Properties

Acyclic

Table[GraphData[{"WheelComplement",n},"Acyclic"],{n,4,20}]

AdjacencyMatrixCount

A171005

seq=Table[GraphData[{"WheelComplement",n},"AdjacencyMatrixCount"],{n,4,20}]

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

AlgebraicConnectivity

seq=Table[GraphData[{"WheelComplement",n},"AlgebraicConnectivity"],{n,4,20}]

AlmostHamiltonian

Table[GraphData[{"WheelComplement",n},"AlmostHamiltonian"],{n,4,20}]

Anarboricity

seq=Table[GraphData[{"WheelComplement",n},"Anarboricity"],{n,4,20}]

Antipodal

Table[GraphData[{"WheelComplement",n},"Antipodal"],{n,4,20}]

Apex

Table[GraphData[{"WheelComplement",n},"Apex"],{n,4,20}]

ApexCount

seq=Table[GraphData[{"WheelComplement",n},"ApexCount"],{n,4,20}]

Arboricity

Table[GraphData[{"WheelComplement",n},"Arboricity"],{n,4,20}]

ArcTransitive

Table[GraphData[{"WheelComplement",n},"ArcTransitive"],{n,4,20}]

ArcTransitivity

Table[GraphData[{"WheelComplement",n},"ArcTransitivity"],{n,4,20}]

ArticulationVertexCount

seq=Table[GraphData[{"WheelComplement",n},"ArticulationVertexCount"],{n,4,20}]

AutomorphismCount

seq=Table[GraphData[{"WheelComplement",n},"AutomorphismCount"],{n,4,20}]

BalabanIndex

seq=Table[GraphData[{"WheelComplement",n},"BalabanIndex"],{n,4,20}]

Bandwidth

Table[GraphData[{"WheelComplement",n},"Bandwidth"],{n,4,20}]

Bicolorable

Table[GraphData[{"WheelComplement",n},"Bicolorable"],{n,4,20}]

Biconnected

Table[GraphData[{"WheelComplement",n},"Biconnected"],{n,4,20}]

Bicubic

Table[GraphData[{"WheelComplement",n},"Bicubic"],{n,4,20}]

Bipartite

Table[GraphData[{"WheelComplement",n},"Bipartite"],{n,4,20}]

BipartiteDimension

seq=Table[GraphData[{"WheelComplement",n},"BipartiteDimension"],{n,4,20}]

Biplanar

Table[GraphData[{"WheelComplement",n},"Biplanar"],{n,4,20}]

BlockCount

seq=Table[GraphData[{"WheelComplement",n},"BlockCount"],{n,4,20}]

BridgeCount

seq=Table[GraphData[{"WheelComplement",n},"BridgeCount"],{n,4,20}]

Bridged

Table[GraphData[{"WheelComplement",n},"Bridged"],{n,4,20}]

BroadcastTime

seq=Table[GraphData[{"WheelComplement",n},"BroadcastTime"],{n,4,20}]

BurningNumber

seq=Table[GraphData[{"WheelComplement",n},"BurningNumber"],{n,4,20}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"CharacteristicPolynomial"],{n,k0=4,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[(((-1-x))^(-1+n)) (-1+n-x),{n,k0,20}]

FullSimplify[poly2-poly]

CheegerConstant

seq=Table[GraphData[{"WheelComplement",n},"CheegerConstant"],{n,4,20}]

Chordal

Table[GraphData[{"WheelComplement",n},"Chordal"],{n,4,20}]

ChordCount

seq=Table[GraphData[{"WheelComplement",n},"ChordCount"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[DeleteMissing@seq,x]

Simplify[SeriesCoefficient[gf,{x,0,n}],n>6]//Factor

ChordlessCycleCount

A027379

seq=Table[GraphData[{"WheelComplement",n},"ChordlessCycleCount"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[DeleteMissing@seq,x]//Simplify

SeriesCoefficient[(x^6) (1+((x (-3+2 x))/(((-1+x))^3))),{x,0,n}]

Simplify[%,n>6]//Factor

Table[(1/2) (-6+n) (-1+n),{n,4,20}]

ChromaticInvariant

seq=Table[GraphData[{"WheelComplement",n},"ChromaticInvariant"],{n,4,20}]

FindSequenceFunction[Drop[seq,2],n-6]

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

ChromaticNumber

seq=Table[GraphData[{"WheelComplement",n},"ChromaticNumber"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[seq,x]

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

FullSimplify[%,n>5]

ChromaticPolynomial

poly=Table[GraphData[{"WheelComplement",n},"ChromaticPolynomial"],{n,4,20}]/.f_Function:>f[x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

CircuitRank

seq=Table[GraphData[{"WheelComplement",n},"CircuitRank"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[seq,x]

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

FullSimplify[%,n>5]//Factor

Circumference

seq=Table[GraphData[{"WheelComplement",n},"Circumference"],{n,4,20}]

Classes

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

ClawFree

Table[GraphData[{"WheelComplement",n},"ClawFree"],{n,4,20}]

CliqueCount

seq=Table[GraphData[{"WheelComplement",n},"CliqueCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

CliqueCoveringNumber

seq=Table[GraphData[{"WheelComplement",n},"CliqueCoveringNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

CliqueNumber

seq=Table[GraphData[{"WheelComplement",n},"CliqueNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

CliquePolynomial

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

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

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

FullSimplify[SeriesCoefficient[gf,{z,0,n}],n>=k0]

Coarseness

seq=Table[GraphData[{"WheelComplement",n},"Coarseness"],{n,4,20}]

CoboundaryPolynomial

(poly=Table[GraphData[{"WheelComplement",n},"CoboundaryPolynomial"],{n,4,20}]/.f_Function:>f[x,y])//Column

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"WheelComplement",n},"ComplementOddChordlessCycleCount"],{n,4,20}]

ComplementOddChordlessCyclePolynomial

Conductance

Connected

Table[GraphData[{"WheelComplement",n},"Connected"],{n,4,20}]

ConnectedComponentCount

Table[GraphData[{"WheelComplement",n},"ConnectedComponentCount"],{n,4,20}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"ConnectedDominatingSetCount"],{n,4,20}]

ConnectedDominationNumber

seq=Table[GraphData[{"WheelComplement",n},"ConnectedDominationNumber"],{n,4,20}]

ConnectedDominationPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"ConnectedDominationPolynomial"],{n,k0=4,20}]/.f_Function:>f[x])

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"WheelComplement",n},"ConnectedInducedSubgraphCount"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[DeleteMissing@seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>5]

-1+x-(-1+n) (x^2) (1+x)+(((1+x))^(-1+n))/.x->1//FullSimplify

ConnectedInducedSubgraphPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=4,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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>5]

Corank

seq=Table[GraphData[{"WheelComplement",n},"Corank"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>5]//Factor

CriticalNonplanar

Table[GraphData[{"WheelComplement",n},"CriticalNonplanar"],{n,4,20}]

CrossingNumber

seq=Table[GraphData[{"WheelComplement",n},"CrossingNumber"],{n,4,20}]

gf=(x^4)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>5]//Factor

Table[(1/128) (821-844 n+286 (n^2)-40 (n^3)+2 (n^4)+(((-1))^n)(331-116n+10(n^2))),{n,4,20}]

Cubic

Table[GraphData[{"WheelComplement",n},"CriticalNonplanar"],{n,4,20}]

CycleCount

seq=Table[GraphData[{"WheelComplement",n},"CycleCount"],{n,4,20}]

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

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

CyclePolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"CyclePolynomial"],{n,k0=4,20}]/.f_Function:>f[x])//Column

(coef=Factor/@FindLinearRecurrence[poly])//Head//Timing

FindGeneratingFunction[poly,z]//Head//Timing

Cyclic

Table[GraphData[{"WheelComplement",n},"Cyclic"],{n,4,20}]

CyclicEdgeConnectivity

Table[GraphData[{"WheelComplement",n},"CyclicEdgeConnectivity"],{n,4,20}]

Degeneracy

seq=Table[GraphData[{"WheelComplement",n},"Degeneracy"],{n,4,20}]

DegreeCentralities

Tally/@Table[GraphData[{"WheelComplement",n},"DegreeCentralities"],{n,4,20}]

DetourIndex

seq=Table[GraphData[{"WheelComplement",n},"DetourIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]//Factor

DetourPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"DetourPolynomial"][x],{n,k0=1,20}])//Column

coef=Factor/@FindLinearRecurrence[poly]

poly2=Table[(((n-1)^2-#) (#+n-1)^(n-1)&)@x,{n,k0,20}]

Abs[FullSimplify[poly2/poly]]

Diameter

seq=Table[GraphData[{"WheelComplement",n},"Diameter"],{n,4,20}]

Dimension

seq=Table[GraphData[{"WheelComplement",n},"Dimension"],{n,4,20}]

DistancePolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"DistancePolynomial"][x],{n,k0=1,20}])//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]

DistanceRegular

Table[GraphData[{"WheelComplement",n},"DistanceRegular"],{n,4,20}]

DistanceTransitive

Table[GraphData[{"WheelComplement",n},"DistanceTransitive"],{n,4,20}]

DistinguishingNumber

seq=Table[GraphData[{"WheelComplement",n},"DistinguishingNumber"],{n,4,20}]

DomaticNumber

seq=Table[GraphData[{"WheelComplement",n},"DomaticNumber"],{n,4,20}]

DominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"DominatingSetCount"],{n,4,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-3]

DominationNumber

seq=Table[GraphData[{"WheelComplement",n},"DominationNumber"],{n,4,20}]

DominationPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"DominationPolynomial"][x],{n,k0=1,20}])//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[-1+(((1+x))^n),{n,k0,20}]

FullSimplify[poly2-poly]

Doublecross

Table[GraphData[{"WheelComplement",n},"Doublecross"],{n,4,20}]

Eccentricities

seq=Table[GraphData[{"WheelComplement",n},"EdgeChromaticNumber"],{n,4,20}]

EccentricityCentralities

seq=Table[GraphData[{"WheelComplement",n},"EdgeChromaticNumber"],{n,4,20}]

EdgeBetweennessCentralities

seq=Table[GraphData[{"WheelComplement",n},"EdgeChromaticNumber"],{n,4,20}]

EdgeChromaticNumber

seq=Table[GraphData[{"WheelComplement",n},"EdgeChromaticNumber"],{n,4,20}]

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

EdgeConnectivity

Table[GraphData[{"WheelComplement",n},"EdgeConnectivity"],{n,4,20}]

EdgeCount

seq=Table[GraphData[{"WheelComplement",n},"EdgeCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

EdgeCoverCount

seq=Table[GraphData[{"WheelComplement",n},"EdgeCoverCount"],{n,4,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-3]

EdgeCoverNumber

seq=Table[GraphData[{"WheelComplement",n},"EdgeCoverNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]

gf=x FindGeneratingFunction[seq,x]

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

EdgeCoverPolynomial

Take[poly=Factor@Table[GraphData[{"WheelComplement",n},"EdgeCoverPolynomial"][x],{n,k0=1,20}],UpTo[3]]//Column

coef=Factor/@FindLinearRecurrence[poly]

FullSimplify[Take[poly2-poly,7]]

EdgeCutCount

seq=Table[GraphData[{"WheelComplement",n},"EdgeCoverNumber"],{n,4,20}]

EdgeTransitive

Table[GraphData[{"WheelComplement",n},"EdgeTransitive"],{n,4,20}]

EigenvectorCentralities

seq=Table[GraphData[{"WheelComplement",n},"EigenvectorCentralities"],{n,4,20}]

EulerianCycleCount

seq=Table[GraphData[{"WheelComplement",n},"EulerianCycleCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

FaceCount

seq=Table[GraphData[{"WheelComplement",n},"FaceCount"],{n,4,20}]

FlowPolynomial

Take[poly=Factor@Table[GraphData[{"WheelComplement",n},"FlowPolynomial"][x],{n,k0=1,20}],UpTo[6]]//Column

coef=Factor/@FindLinearRecurrence[poly]

FractionalArboricity

seq=Table[GraphData[{"WheelComplement",n},"FractionalArboricity"],{n,4,20}]

FractionalChromaticNumber

seq=Table[GraphData[{"WheelComplement",n},"FractionalChromaticNumber"],{n,4,20}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"WheelComplement",n},"FractionalEdgeChromaticNumber"],{n,4,20}]

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

Genus

seq=Table[GraphData[{"WheelComplement",n},"Genus"],{n,4,20}]

Geodetic

Table[GraphData[{"WheelComplement",n},"Geodetic"],{n,4,20}]

Girth

seq=Table[GraphData[{"WheelComplement",n},"Girth"],{n,4,20}]

GlobalClusteringCoefficient

seq=Table[GraphData[{"WheelComplement",n},"GlobalClusteringCoefficient"],{n,4,20}]

Gonality

seq=Table[GraphData[{"WheelComplement",n},"Gonality"],{n,4,20}]

Graceful

Table[GraphData[{"WheelComplement",n},"Graceful"],{n,4,20}]

GracefulLabelingCount

seq=Table[GraphData[{"WheelComplement",n},"GracefulLabelingCount"],{n,4,20}]

HadwigerNumber

Table[GraphData[{"WheelComplement",n},"HadwigerNumber"],{n,4,20}]

HamiltonConnected

Table[GraphData[{"WheelComplement",n},"HamiltonConnected"],{n,4,20}]

HamiltonDecomposable

Table[GraphData[{"WheelComplement",n},"HamiltonDecomposable"],{n,4,20}]

HamiltonDecompositionCount

seq=Table[GraphData[{"WheelComplement",n},"HamiltonDecompositionCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

Hamiltonian

Table[GraphData[{"WheelComplement",n},"Hamiltonian"],{n,4,20}]

HamiltonianCycleCount

seq=Table[GraphData[{"WheelComplement",n},"HamiltonianCycleCount"],{n,4,20}]

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

HamiltonianNumber

seq=Table[GraphData[{"WheelComplement",n},"HamiltonianNumber"],{n,4,20}]

HamiltonianPathCount

seq=Table[GraphData[{"WheelComplement",n},"HamiltonianPathCount"],{n,4,20}]

FindSequenceFunction[Drop[seq,1],n-1]

HamiltonianWalkCount

seq=Table[GraphData[{"WheelComplement",n},"HamiltonianWalkCount"],{n,4,20}]

HamiltonLaceable

Table[GraphData[{"WheelComplement",n},"HamiltonianWalkCount"],{n,4,20}]

HararyIndex

seq=Table[GraphData[{"WheelComplement",n},"HararyIndex"],{n,4,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-3]

HexagonCount

seq=Table[GraphData[{"WheelComplement",n},"HexagonCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

HosoyaIndex

seq=Table[GraphData[{"WheelComplement",n},"HosoyaIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]

(x^3)FindGeneratingFunction[seq,x]

FindLinearRecurrence[seq]

HStarConnected

Table[GraphData[{"WheelComplement",n},"HStarConnected"],{n,4,20}]

Hypohamiltonian

Table[GraphData[{"WheelComplement",n},"Hypohamiltonian"],{n,4,20}]

Hypotraceable

Table[GraphData[{"WheelComplement",n},"Hypotraceable"],{n,4,20}]

Identity

Table[GraphData[{"WheelComplement",n},"Identity"],{n,4,20}]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"IdiosyncraticPolynomial"][x,y],{n,k0=1,10}])//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[(x+1)^(n-1) (x+1-n),{n,k0,10}]

FullSimplify[poly2/poly]

IndependenceNumber

seq=Table[GraphData[{"WheelComplement",n},"IndependenceNumber"],{n,4,20}]

IndependencePolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"IndependencePolynomial"],{n,k0=1,20}]/.f_Funtion:>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[n x+1,{n,l0=10}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"WheelComplement",n},"IndependenceRatio"],{n,4,20}]

FindSequenceFunction[seq,n-3]

IndependentEdgeSetCount

seq=Table[GraphData[{"WheelComplement",n},"IndependentEdgeSetCount"],{n,4,20}]

Table[(I/Sqrt[2])^n HermiteH[n,-I/Sqrt[2]],{n,4,20}]

IndependentVertexSetCount

seq=Table[GraphData[{"WheelComplement",n},"IndependentVertexSetCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

IntersectionNumber

seq=Table[GraphData[{"WheelComplement",n},"IntersectionNumber"],{n,4,20}]

IrredundanceNumber

seq=Table[GraphData[{"WheelComplement",n},"IrredundanceNumber"],{n,4,20}]

IrredundancePolynomial

poly=Table[GraphData[{"WheelComplement",n},"IrredundanceNumber"],{n,4,20}]/.f_Function:>f[x]

IrredundantSetCount

seq=Table[GraphData[{"WheelComplement",n},"IrredundantSetCount"],{n,4,20}]

IsolatedPointCount

seq=Table[GraphData[{"WheelComplement",n},"IsolatedPointCount"],{n,4,20}]

KatzCentralities

seq=Table[GraphData[{"WheelComplement",n},"KatzCentralities"],{n,4,20}]

KirchhoffIndex

seq=Table[GraphData[{"WheelComplement",n},"KirchhoffIndex"],{n,4,20}]

KirchhoffSumIndex

seq=Table[GraphData[{"WheelComplement",n},"KirchhoffSumIndex"],{n,4,20}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"LaplacianPolynomial"][x],{n,k0=1,10}])//Column

LeafCount

seq=Table[GraphData[{"WheelComplement",n},"LeafCount"],{n,4,20}]

Likelihood

seq=Table[GraphData[{"WheelComplement",n},"Likelihood"],{n,4,20}]

LinearArboricity

seq=Table[GraphData[{"WheelComplement",n},"LinearArboricity"],{n,4,20}]

LocalClusteringCoefficients

seq=Table[GraphData[{"WheelComplement",n},"LocalClusteringCoefficients"],{n,4,20}]

LongestCycleCount

seq=Table[GraphData[{"WheelComplement",n},"LongestCycleCount"],{n,4,20}]

LongestPathCount

seq=Table[GraphData[{"WheelComplement",n},"LongestPathCount"],{n,4,20}]

LongestPathLength

seq=Table[GraphData[{"WheelComplement",n},"LongestPathLength"],{n,4,20}]

LovaszNumber

seq=Table[GraphData[{"WheelComplement",n},"LovaszNumber"],{n,4,20}]

LowerIndependenceNumber

seq=Table[GraphData[{"WheelComplement",n},"LowerIndependenceNumber"],{n,4,20}]

LowerMatchingNumber

seq=Table[GraphData[{"WheelComplement",n},"LowerMatchingNumber"],{n,4,20}]

MatchingGeneratingPolynomial

Take[poly=Factor@Table[GraphData[{"WheelComplement",n},"MatchingGeneratingPolynomial"][x],{n,k0=1,20}],6]//Column

coef=Factor/@FindLinearRecurrence[poly]

FullSimplify[poly2-poly]

MatchingNumber

seq=Table[GraphData[{"WheelComplement",n},"MatchingNumber"],{n,4,20}]

Table[Floor[n/2],{n,4,20}]

MatchingPolynomial

Take[poly=Factor@Table[GraphData[{"WheelComplement",n},"MatchingPolynomial"][x],{n,k0=1,20}],6]//Column

coef=Factor/@FindLinearRecurrence[poly]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"WheelComplement",n},"MaximalCliqueCount"],{n,4,20}]

MaximalCliquePolynomial

poly=Table[GraphData[{"WheelComplement",n},"MaximalCliqueCount"],{n,4,20}]/.f_Function:>f[x]

MaximalIndependencePolynomial

(poly=Table[GraphData[{"WheelComplement",n},"MaximalIndependencePolynomial"],{n,4,20}]/.f_Function:>f[x])//Column

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximalIndependentEdgeSetCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximalIndependentVertexSetCount"],{n,4,20}]

MaximalIrredundancePolynomial

poly=Table[GraphData[{"WheelComplement",n},"MaximalIrredundancePolynomial"],{n,4,20}]/.f_Function:>f[x]

MaximalIrredundantSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximalIrredundantSetCount"],{n,4,20}]

MaximalMatchingGeneratingPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MaximalMatchingGeneratingPolynomial"],{n,4,20}]/.f_Function:>f[x]

MaximumCliqueCount

seq=Table[GraphData[{"WheelComplement",n},"MaximumCliqueCount"],{n,4,20}]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximumIndependentEdgeSetCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximumIndependentVertexSetCount"],{n,4,20}]

MaximumIrredundantSetCount

seq=Table[GraphData[{"WheelComplement",n},"MaximumIrredundantSetCount"],{n,4,20}]

MaximumLeafNumber

seq=Table[GraphData[{"WheelComplement",n},"MaximumLeafNumber"],{n,4,20}]

MaximumVertexDegree

seq=Table[GraphData[{"WheelComplement",n},"MaximumVertexDegree"],{n,4,20}]

MeanClusteringCoefficient

seq=Table[GraphData[{"WheelComplement",n},"MeanClusteringCoefficient"],{n,4,20}]

MeanDistance

seq=Table[GraphData[{"WheelComplement",n},"MeanDistance"],{n,4,20}]

FindSequenceFunction[seq,n-3]

Median

seq=Table[GraphData[{"WheelComplement",n},"Median"],{n,4,20}]

MetricDimension

seq=Table[GraphData[{"WheelComplement",n},"MetricDimension"],{n,4,20}]

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalConnectedDominatingSetCount"],{n,4,20}]

MinimalConnectedDominationPolynomial

seq=Table[GraphData[{"WheelComplement",n},"MinimalConnectedDominationPolynomial"],{n,4,20}]

MinimalDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalDominatingSetCount"],{n,4,20}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"WheelComplement",n},"MinimalDominatingSetSignature"],{n,4,20}]

MinimalDominationPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MaximalMatchingGeneratingPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimalEdgeCoverCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalEdgeCoverCount"],{n,4,20}]

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

FindLinearRecurrence[seq]

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

MinimalEdgeCoverPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MinimalEdgeCoverPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimalEdgeCutCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalEdgeCutCount"],{n,4,20}]

MinimalEdgeCutPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MinimalEdgeCutPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalTotalDominatingSetCount"],{n,4,20}]

MinimalTotalDominationPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MinimalTotalDominationPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimalVertexCoverCount

seq=Table[GraphData[{"WheelComplement",n},"MinimalVertexCoverCount"],{n,4,20}]

MinimalVertexCoverPolynomial

poly=Table[GraphData[{"WheelComplement",n},"MinimalVertexCoverPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimalVertexCutCount

Inferred for n > 16:

seq=Table[GraphData[{"WheelComplement",n},"MinimalVertexCutCount"],{n,4,20}]

MinimalVertexCutPolynomial

Inferred for n > 16:

poly=Table[GraphData[{"WheelComplement",n},"MinimalVertexCutPolynomial"],{n,4,20}]/.f_Function:>f[x]

MinimumCliqueCoveringCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumCliqueCoveringCount"],{n,4,20}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumConnectedDominatingSetCount"],{n,4,20}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumCoveringsByMaximalCliquesCount"],{n,4,20}]

MinimumCyclicEdgeCutCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumCyclicEdgeCutCount"],{n,4,20}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumDistinguishingLabelingCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

MinimumDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumDominatingSetCount"],{n,4,20}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumEdgeCoverCount"],{n,4,20}]

MinimumEdgeCutCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumEdgeCutCount"],{n,4,20}]

MinimumLeafNumber

seq=Table[GraphData[{"WheelComplement",n},"MinimumLeafNumber"],{n,4,20}]

MinimumPathCoveringCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumPathCoveringCount"],{n,4,20}]

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumTotalDominatingSetCount"],{n,4,20}]

MinimumVertexCoverCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumVertexCoverCount"],{n,4,20}]

MinimumVertexCutCount

seq=Table[GraphData[{"WheelComplement",n},"MinimumVertexCutCount"],{n,4,20}]

MinimumVertexDegree

seq=Table[GraphData[{"WheelComplement",n},"MinimumVertexDegree"],{n,4,20}]

MolecularTopologicalIndex

seq=Table[GraphData[{"WheelComplement",n},"MolecularTopologicalIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

Nonhamiltonian

Table[GraphData[{"WheelComplement",n},"Nonhamiltonian"],{n,4,20}]

NoPerfectMatching

Table[GraphData[{"WheelComplement",n},"NoPerfectMatching"],{n,4,20}]

OddChordlessCycleCount

seq=Table[GraphData[{"WheelComplement",n},"OddChordlessCycleCount"],{n,4,20}]

OddChordlessCyclePolynomial

poly=Table[GraphData[{"WheelComplement",n},"OddChordlessCyclePolynomial"],{n,4,20}]/.f_Function:>f[x]

Outerplanar

Table[GraphData[{"WheelComplement",n},"Outerplanar"],{n,4,20}]

PageRankCentralities

PathCount

seq=Table[GraphData[{"WheelComplement",n},"PathCount"],{n,4,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

PathCoveringNumber

seq=Table[GraphData[{"WheelComplement",n},"PathCoveringNumber"],{n,4,20}]

PathPolynomial

(poly=Factor[Table[GraphData[{"WheelComplement",n},"PathPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,7],x]

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

PentagonCount

seq=Table[GraphData[{"WheelComplement",n},"PentagonCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

Perfect

Table[GraphData[{"WheelComplement",n},"Perfect"],{n,4,20}]

PerfectMatching

Table[GraphData[{"WheelComplement",n},"PerfectMatching"],{n,4,20}]

Periphery

seq=Table[GraphData[{"WheelComplement",n},"Periphery"],{n,4,20}]

Planar

Table[GraphData[{"WheelComplement",n},"Planar"],{n,4,20}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"WheelComplement",n},"ProjectivePlaneCrossingNumber"],{n,4,20}]

Pseudoarboricity

seq=Table[GraphData[{"WheelComplement",n},"Pseudoarboricity"],{n,4,20}]

QChromaticPolynomial

(poly=Factor[Table[GraphData[{"WheelComplement",n},"QChromaticPolynomial"],{n,k0=1,kmax=20}]/.f_Function:>f[x]])//Column

RadialityCentralities

seq=Table[GraphData[{"WheelComplement",n},"RadialityCentralities"],{n,4,20}]

RadioLabelingCount

seq=Table[GraphData[{"WheelComplement",n},"RadioLabelingCount"],{n,4,20}]

RadioNumber

seq=Table[GraphData[{"WheelComplement",n},"RadioNumber"],{n,4,20}]

Radius

seq=Table[GraphData[{"WheelComplement",n},"Radius"],{n,4,20}]

Rank

seq=Table[GraphData[{"WheelComplement",n},"Rank"],{n,4,20}]

RankPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"RankPolynomial"],{n,k0=4,20}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"WheelComplement",n},"RectilinearCrossingNumber"],{n,4,20}]

FindSequenceFunction[DeleteMissing@seq,n]

Regular

Table[GraphData[{"WheelComplement",n},"Regular"],{n,4,20}]

RegularParameters

seq=Table[GraphData[{"WheelComplement",n},"RegularParameters"],{n,4,20}]

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"ReliabilityPolynomial"][x],{n,k0=1,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ResistanceMatrix

MatrixForm/@Table[GraphData[{"WheelComplement",n},"ResistanceMatrix"],{n,6}]

ScrambleNumber

seq=Table[GraphData[{"WheelComplement",n},"ScrambleNumber"],{n,4,20}]

ShannonCapacity

seq=Table[GraphData[{"WheelComplement",n},"ShannonCapacity"],{n,4,20}]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"SigmaPolynomial"][x],{n,k0=1,10}])//Column

coef=Factor/@FindLinearRecurrence[poly]

coef=Factor/@FindPolynomialRecurrence[poly,6]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[%-poly]

Singlecross

Table[GraphData[{"WheelComplement",n},"Singlecross"],{n,4,20}]

Skewness

seq=Table[GraphData[{"WheelComplement",n},"Skewness"],{n,4,20}]

SpanningTreeCount

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

FindSequenceFunction[seq,n-3]

SpectralRadius

seq=Table[GraphData[{"WheelComplement",n},"SpectralRadius"],{n,4,20}]

SquareCount

seq=Table[GraphData[{"WheelComplement",n},"SquareCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

SquareFree

Table[GraphData[{"WheelComplement",n},"SquareFree"],{n,4,20}]

StabilityIndex

seq=Table[GraphData[{"WheelComplement",n},"StabilityIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]

gf=x FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>1]

StarArboricity

seq=Table[GraphData[{"WheelComplement",n},"StarArboricity"],{n,4,20}]

StatusCentralities

seq=Table[GraphData[{"WheelComplement",n},"StatusCentralities"],{n,4,20}]

Strength

seq=Table[GraphData[{"WheelComplement",n},"Strength"],{n,4,20}]

StronglyPerfect

Table[GraphData[{"WheelComplement",n},"StronglyPerfect"],{n,4,20}]

StronglyRegular

Table[GraphData[{"WheelComplement",n},"StronglyRegular"],{n,4,20}]

Symmetric

Table[GraphData[{"WheelComplement",n},"Symmetric"],{n,4,20}]

SymmetricallyDistinctVertexPairCount

seq=Table[GraphData[{"WheelComplement",n},"SymmetricallyDistinctVertexPairCount"],{n,4,20}]

SymmetricallyDistinctVertexPairSignature

seq=Table[GraphData[{"WheelComplement",n},"SymmetricallyDistinctVertexPairSignature"],{n,4,20}]

Thickness

seq=Table[GraphData[{"WheelComplement",n},"Thickness"],{n,4,20}]

TopologicalIndex

seq=Table[GraphData[{"WheelComplement",n},"TopologicalIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]//FullSimplify

ToroidalCrossingNumber

seq=Table[GraphData[{"WheelComplement",n},"ToroidalCrossingNumber"],{n,4,20}]

TotalDominatingSetCount

seq=Table[GraphData[{"WheelComplement",n},"TotalDominatingSetCount"],{n,4,20}]

TotalDominationNumber

seq=Table[GraphData[{"WheelComplement",n},"TotalDominationNumber"],{n,4,20}]

TotalDominationPolynomial

poly=Table[GraphData[{"WheelComplement",n},"TotalDominationNumber"],{n,4,20}]/.f_Function:>f[x]

Toughness

seq=Table[GraphData[{"WheelComplement",n},"Toughness"],{n,4,20}]

Traceable

Table[GraphData[{"WheelComplement",n},"Traceable"],{n,4,20}]

TreeDepth

seq=Table[GraphData[{"WheelComplement",n},"TreeDepth"],{n,4,20}]

TreeNumber

seq=Table[GraphData[{"WheelComplement",n},"TreeNumber"],{n,4,20}]

Treewidth

seq=Table[GraphData[{"WheelComplement",n},"Treewidth"],{n,4,20}]

Triameter

seq=Table[GraphData[{"WheelComplement",n},"Triameter"],{n,4,20}]

TriangleCount

seq=Table[GraphData[{"WheelComplement",n},"TriangleCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

TriangleFree

Table[GraphData[{"WheelComplement",n},"TriangleFree"],{n,4,20}]

TuttePolynomial

(Table[GraphData[{"WheelComplement",n},"TuttePolynomial"],{n,4,20}]/.f_Function:>f[x,y])//Column

Unicyclic

Table[GraphData[{"WheelComplement",n}],{n,4,8}]

seq=Table[GraphData[{"WheelComplement",n},"Unicyclic"],{n,4,20}]

UniquelyColorable

seq=Table[GraphData[{"WheelComplement",n},"UniquelyColorable"],{n,4,20}]

UniquelyPancyclic

seq=Table[GraphData[{"WheelComplement",n},"UniquelyPancyclic"],{n,4,20}]

UnitDistance

seq=Table[GraphData[{"WheelComplement",n},"UnitDistance"],{n,4,20}]

UpperDominationNumber

seq=Table[GraphData[{"WheelComplement",n},"UpperDominationNumber"],{n,4,20}]

UpperIrredundanceNumber

seq=Table[GraphData[{"WheelComplement",n},"UpperIrredundanceNumber"],{n,4,20}]

VertexConnectivity

Table[GraphData[{"WheelComplement",n},"VertexConnectivity"],{n,4,20}]

VertexCount

seq=Table[GraphData[{"WheelComplement",n},"VertexCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

VertexCoverCount

seq=Table[GraphData[{"WheelComplement",n},"VertexCoverCount"],{n,4,20}]

FindSequenceFunction[seq,n-3]

x FindGeneratingFunction[DeleteMissing@seq,x]

VertexCoverNumber

seq=Table[GraphData[{"WheelComplement",n},"VertexCoverNumber"],{n,4,20}]

FindSequenceFunction[seq,n-3]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"WheelComplement",n},"VertexCoverPolynomial"][x],{n,k0=1,10}])//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[(x^(n-1)) (x+n),{n,10}]

FullSimplify[poly2-poly]

VertexCutCount

seq=Table[GraphData[{"WheelComplement",n},"VertexCutCount"],{n,4,20}]

VertexDegrees

Tally/@Table[GraphData[{"WheelComplement",n},"VertexDegrees"],{n,4,20}]

VertexTransitive

seq=Table[GraphData[{"WheelComplement",n},"VertexTransitive"],{n,4,20}]

And@@seq

WienerIndex

seq=Table[GraphData[{"WheelComplement",n},"WienerIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]

WienerSumIndex

seq=Table[GraphData[{"WheelComplement",n},"WienerSumIndex"],{n,4,20}]

FindSequenceFunction[seq,n-3]

Embeddings

<<MathWorld`Graphs`

{"WheelComplement", 9}

GraphData[{"WheelComplement",9},"StandardName"]

All

GraphData[{"WheelComplement",9},"Graph","All"]//StyleGraphs

Halin

GraphData[{"WheelComplement",9},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"WheelComplement",9},"LCFSignature"]

GraphData[{"WheelComplement",9},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"WheelComplement",9},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"WheelComplement",9}],Debug->True]//Timing

Matchstick

GraphData[{"WheelComplement",9},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"WheelComplement",9},{"RCN","CN"}]

GraphData[{"WheelComplement",9},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot[Graph[GraphData[{"WheelComplement",9},"VertexList"],GraphData[{"WheelComplement",9},"Edges"]]]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"WheelComplement",9}],"CrossingNumber"->2]]

g0=GraphData[GraphData[{"WheelComplement",9},"ConnectedComponents","Name"][[1]],"Graph","MinimalCrossing"][[1]]

v=GraphEmbedding[g0];

g=Graph[Range[Length[v]+1],EdgeList[g0],VertexCoordinates->Append[v,{0,-1/10}+Min/@Transpose[v]]]

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"WheelComplement",9},"Edges"]]

3D All

GraphData[{"WheelComplement",9},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"WheelComplement",9},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"WheelComplement",9},"Graph",{"3D","Polyhedron"}]

{"WheelComplement", 10}

GraphData[{"WheelComplement", 10},"StandardName"]

Primary

GraphData[{"WheelComplement", 10}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"WheelComplement", 10},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"WheelComplement", 10},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"WheelComplement", 10},"EdgeList"]]

Default embedding

GraphPlot@RotateGraph[Graph@GraphData[{"WheelComplement", 10},"Edges"],Pi/2]

v=Join[UnitCirclePoints[7],{#/2,0}&/@{-1,1}];
AppendTo[v,v[[1]]+{3/4,0}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,1,9,4,8,7,9,3,5,8,1,3,8,6,4,2,6,1],e[7,2,9,5,7]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

Embddings from connected components

cc=GraphData[{"WheelComplement", 10},"ConnectedComponents","Name"]

GraphData[cc[[1]],"RCN"]

GraphData[cc[[1]],"Graph","MinimalCrossing"]

GraphData[cc[[1]],"Graph","All"]

CirculantIndices[GraphData[cc[[1]],"Graph","Circulant"][[1]],All]

gs=CirculantGraph1@@@%

GraphData[cc[[1]],"Graph","LCF"]

LCFFromEmbedding[]

LCFGraph@@%

gs0={,,,};

gs=MyGraphUnion[#,Graph[{1},{}],Offset->1/10]&/@gs0

Degenerate

GraphData[{"WheelComplement", 10},"Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData[{"WheelComplement", 10},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"WheelComplement", 10},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"WheelComplement", 10},"Planar"]

GraphData[{"WheelComplement", 10},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"WheelComplement", 10},{"RCN","CN"}]

GraphData[{"WheelComplement", 10},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"WheelComplement", 10}],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"WheelComplement", 10}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"WheelComplement", 10},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"WheelComplement", 10}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

cc=GraphData[{"WheelComplement", 10},"ConnectedComponents","Name"]

GraphData[#,"Graph","MinimalCrossing"]&/@cc

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"WheelComplement", 10},"Circulant"]

GraphData[{"WheelComplement", 10},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"WheelComplement", 10},"Halin"]

GraphData[{"WheelComplement", 10},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"WheelComplement", 10},"LCF"]

GraphData[{"WheelComplement", 10},"HamiltonianCycleCount"]

GraphData[{"WheelComplement", 10},"LCFSignature"]

GraphData[{"WheelComplement", 10},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"WheelComplement", 10},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"WheelComplement", 10}],Debug->True]//Timing

GraphData[{"WheelComplement", 10},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"WheelComplement", 10},"VertexList"],GraphData[{"WheelComplement", 10},"EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData[{"WheelComplement", 10},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"WheelComplement", 10}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

Matchstick

GraphData[{"WheelComplement", 10},"Matchstick"]

GraphData[{"WheelComplement", 10},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"WheelComplement", 10},"Edges"]]

3D All

GraphData[{"WheelComplement", 10},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"WheelComplement", 10},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"WheelComplement", 10},"Polyhedral"]

GraphData[{"WheelComplement", 10},"Graph",{"3D","Polyhedron"}]

GraphData

Existing

new=Cases[data,{n_,_,gn_?DataPaclets`GraphDataDump`MyGraphQ,__}:>Rule[gn,n]]

AddIndexedClass["WheelComplement",new,"13.1"]

RenameGraphFiles[]

GraphDataString

file=OpenAppend["~/wheel-complement.m"]

Do[WriteString[file,GraphDataString[WheelComplementGraph[n],
"StandardName"->{"WheelComplement",n},
"Name"->ToString[n]<>"\[Hyphen]wheel complement graph",
"Notation"->Subscript[OverBar["W"],n],
"Information"->"WheelComplementGraph",
"Classes"->{"WheelComplement"},
"NotationRules"->{"WheelComplement"->n},
"HadwigerNumber"->Floor[3(n-1)/4],
"Skip"->{"ChromaticPolynomial","TuttePolynomial"}
]],{n,19,20}]