Book Graph

Author

Eric W. Weisstein
December 28, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Graphs.m

GraphicsRow[Table[Show[BookGraph[n]//StyleGraphs,PlotLabel->Style[Subscript[B,n],FontFamily->"Times"]],{n,6}],ImageSize->500,Dividers->All]

GraphData

GraphicsRow[Table[Show[GraphData[{"Book",n}]//StyleGraphs,PlotLabel->Style[Subscript[B,n],FontFamily->"Times"]],{n,6}],ImageSize->500,Dividers->All]

Combinatorica

GraphicsRow[CombinatoricaGraphToV8Graph[#,VertexSize->{"Scaled",.07},System`VertexStyle->Red,System`EdgeStyle->Black]&/@Table[GraphProduct[Star[n+1],Path[n]],{n,6}],Dividers->All,ImageSize->500]

Embeddings

2D

BookGraph[6,VertexLabels->"Name"]

BookGraph[6,VertexCoordinates->Automatic,VertexLabels->"Name"]

System`Graph[EdgeList[BookGraph[6]],VertexLabels->"Name"]

Table[GraphData[{"Book",n},"Graph",{"All","Labeled"}],{n,2,20}]

Table[System`Graph[EdgeList[GraphData[{"Book",n},"Graph"]]],{n,2,20}]

UnitDistance

Table[GraphData[{"Book",n},"Graph","UnitDistance"],{n,20}]

Table[GraphData[{"Book",n},"Graph","UnitDistance"],{n,20}]

3D

Graph3D[BookGraph[6,VertexCoordinates->Automatic,VertexLabels->"Name"]]

Special cases

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

Isomorphism

ToEntity/@BookGraph/@Range[20]

Grid[DeleteCases[Table[{n,RecognizeGraph[BookGraph[n]]},{n,20}],{_,{"Book",_}}],Dividers->All]

Grid[{#,GraphData[#,"StandardNames"]}&/@{"SquareGraph","DominoGraph",{"Book",3},{"Book",4}},Dividers->All,Alignment->Left]

Grid[{#,GraphData[#,"Names"]}&/@{"SquareGraph","DominoGraph",{"Book",3},{"Book",4}},Dividers->All,Alignment->Left]

Properties

Acyclic

seq=Table[GraphData[{"Book",n},"Acyclic"],{n,20}]

AdjacencyMatrixCount

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

FullSimplify[(((2(n+1))!)/(2n!))]

Table[(((2(n+1))!)/(Piecewise[{{8,n==1}},2n!])),{n,20}]

Arboricity

Table[GraphData[{"Book",n},"Arboricity"],{n,20}]

ArcTransitive

seq=Table[GraphData[{"Book",n},"ArcTransitive"],{n,20}]

ArcTransitivity

seq=Table[GraphData[{"Book",n},"ArcTransitivity"],{n,20}]

AutomorphismCount

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

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

BalabanIndex

seq=Table[GraphData[{"Book",n},"BalabanIndex"],{n,4}]//FullSimplify

Bandwidth

seq=Table[GraphData[{"Book",n},"Bandwidth"],{n,20}]

BridgeCount

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

BurningNumber

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

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"CharacteristicPolynomial"],{n,k0=1,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]

Chordal

seq=Table[GraphData[{"Book",n},"Chordal"],{n,20}]

ChordCount

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

Chordless

seq=Table[GraphData[{"Book",n},"Chordless"],{n,20}]

ChordlessCycleCount

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

Chords

seq=Table[GraphData[{"Book",n},"Chords"],{n,20}]

ChromaticallyUnique

seq=Table[GraphData[{"Book",n},"ChromaticallyUnique"],{n,20}]

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

ChromaticInvariant

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

ChromaticNumber

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

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"ChromaticPolynomial"],{n,k0=1,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]

Circumference

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

Classes

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

CliqueCount

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

FindSequenceFunction[seq,n]

CliqueCoveringNumber

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

FindSequenceFunction[seq,n]

CliqueNumber

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

CliquePolynomial

(poly=Factor[Table[GraphData[{"Book",n},"CliquePolynomial"],{n,k0=1,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]

CochromaticGraphNames

Table[GraphData[{"Book",n},"CochromaticGraphNames"],{n,20}]//Column

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

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

A242985(n) for n > 1

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

FindLinearRecurrence[DeleteCases[seq,_Missing]]

FindLinearRecurrence[DeleteCases[Drop[seq,1],_Missing]]

LinearRecurrence[{6,-8},{8,24},{0,20}]

seq=Table[(4^n)+(2^(n+1)),{n,0,20}]

gf=x FindGeneratingFunction[seq,x]//Factor

CoefficientList[Series[-(((-3+10 x))/((-1+2 x) (-1+4 x))),{x,0,20}],x]

FindSequenceFunction[DeleteCases[Drop[seq,1],_Missing],n-1]//FullSimplify

Table[Piecewise[{{9,n==1}},(2^(1+n))+(4^n)],{n,20}]

FullSimplify[If[n==1,1,0]+(((1+x))^n) (2 (x^(n-1))+(((1+x))^n))/.x->1]

ConnectedDominationNumber

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

ConnectedDominationPolynomial

related to A213669

(poly=Factor/@Table[GraphData[{"Book",n},"ConnectedDominationPolynomial"],{n,k0=2,kmax=10}]/.f_Function:>f[x])//Column

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

CoefficientList[Take[DeleteMissing[poly],6],x]

p=.

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[Piecewise[{{4 (x^2)+4 (x^3)+(x^4),n==1}},x (x (((1+x))^(2 n))+2 (((x (1+x)))^n))],{n,2,10}]

FullSimplify[poly2-poly]

CoefficientList[Take[poly2,5],x]

CoefficientList[Table[Piecewise[{{4 +4 x+(x^2),n==1}},(((1+x))^n) (2 (x^(n-1))+(((1+x))^n))],{n,5}],x]

CoefficientList[Table[If[n==1,1,0]+(((1+x))^n) (2 (x^(n-1))+(((1+x))^n)),{n,5}],x]

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

ConnectedInducedSubgraphCount

A290720

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

FindLinearRecurrence[seq]

seq=LinearRecurrence[{9,-27,31,-12},{13,40,127,430},{0,40}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]//FullSimplify

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

Table[2 (3^n)+(4^n)+3 n,{n,0,20}]

gf=FindGeneratingFunction[seq,x]

Series[((3-14 x+4 (x^2)+25 (x^3))/((((-1+x))^2) (1-7 x+12 (x^2)))),{x,0,20}]

CoefficientList[Series[((3-14 x+4 (x^2)+25 (x^3))/((((-1+x))^2) (1-7 x+12 (x^2)))),{x,0,20}],x]

ConnectedInducedSubgraphPolynomial

(poly=DeleteCases[Factor@Table[GraphData[{"Book",n},"ConnectedInducedSubgraphPolynomial"][x],{n,k0=1,kmax=20}],_Missing[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[x (x (((1+x))^(2 n))+n (2+x)+2 (((1+x+(x^2)))^n)),{n,Length[poly]}]

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

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

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

CyclePolynomial

(poly=Factor[Table[GraphData[{"Book",n},"CyclePolynomial"],{n,k0=1,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]

Cyclic

Table[GraphData[{"Book",n},"Cyclic"],{n,20}]

CyclomaticNumber

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

FindSequenceFunction[seq,n]

DetourIndex

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

FindSequenceFunction[seq,n]

gf=x FindSequenceFunction[seq,x]

DetourMatrix

MatrixForm/@Table[GraphData[{"Book",n},"DetourMatrix"],{n,20}]

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

DetourPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"DetourPolynomial"],{n,k0=1,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]

Diameter

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

DistanceMatrix

MatrixForm/@Table[GraphData[{"Book",n},"DistanceMatrix"],{n,20}]

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

DistancePolynomial

(poly=Factor[Table[GraphData[{"Book",n},"DistancePolynomial"],{n,k0=1,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[{"Book",n},"DistinguishingNumber"],{n,20}]

DomaticNumber

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

DominatingSetCount

A288795

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

FindSequenceFunction[seq,n]//FullSimplify

seq=Table[4^n+3^(n+1)-2,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{8,-19,12},{11,41,143},20]

gf=x FindGeneratingFunction[seq,x]//Factor

ToString[gf,InputForm]

CoefficientList[Series[-(((11-47 x+24 (x^2)))/((-1+x) (-1+3 x) (-1+4 x))),{x,0,20}],x]

DominationNumber

seq=Table[GraphData[{"Book",n},"DominationNumber"],{n,10}]

DominationPolynomial

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

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

CoefficientList[Take[DeleteMissing[poly],6],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]

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

FullSimplify[Take[poly2-poly,Length[DeleteMissing[poly]]]]

CoefficientList[Take[poly2,5],x]

CoefficientList[Table[Piecewise[{{4 +4 x+(x^2),n==1}},(((1+x))^n) (2 (x^(n-1))+(((1+x))^n))],{n,5}],x]

?KroneckerDelta

CoefficientList[Table[If[n==1,1,0]+(((1+x))^n) (2 (x^(n-1))+(((1+x))^n)),{n,5}],x]

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

EdgeChromaticNumber

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

FindSequenceFunction[seq,n]

EdgeConnectivity

Table[GraphData[{"Book",n},"EdgeConnectivity"],{n,20}]

EdgeCount

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

FindSequenceFunction[seq,n]

EdgeCoverCount

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

FindSequenceFunction[seq,n]//FullSimplify

EdgeCoverNumber

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

FindSequenceFunction[seq,n]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"EdgeCoverPolynomial"],{n,k0=1,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[{"Book",n},"EulerianCycleCount"],{n,20}]

FaceCount

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

FlowPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"FlowPolynomial"],{n,k0=1,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]

FractionalChromaticNumber

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

FractionalEdgeChromaticNumber

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

FindSequenceFunction[seq,n]

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

Table[GraphData[{"Book",n},"Hamiltonian"],{n,20}]

HamiltonianCycleCount

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

HamiltonianNumber

Probably 2(n+1)

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

HamiltonianPathCount

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

HamiltonianWalkCount

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

HararyIndex

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

FindSequenceFunction[seq,n]

HexagonCount

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

FindSequenceFunction[seq,n]

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"IdiosyncraticPolynomial"][x,y],{n,k0=1,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]

IndependenceNumber

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

FindSequenceFunction[seq,n]

IndependencePolynomial

(poly=Factor[Table[GraphData[{"Book",n},"IndependencePolynomial"],{n,k0=1,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]

IndependenceRatio

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

IndependentEdgeSetCount

A084264

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

FindSequenceFunction[seq,n-1]//FullSimplify//Factor

Table[(2^(n-3)) (4+3 n+(n^2)),{n,0,20}]

seq=Table[If[n==0,1,(2^(n-3))((n^2)+3n+4)],{n,0,20}]

FindLinearRecurrence[seq]

Join[{1},LinearRecurrence[{6,-12,8},{2,7,22},20]]

FindGeneratingFunction[seq,x]

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

IndependentVertexSetCount

A085279

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

FindLinearRecurrence[seq]

Join[{1},LinearRecurrence[{5,-6},{3,7},20]]

FindSequenceFunction[seq,n]

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

IntersectionNumber

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

IrredundantSetCount

A290651

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

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Table[5-(2^(1+n))+(3^(1+n))+n,{n,-1,40}]

seq=Table[5-(2^(n-1))+(3^(n-1))+n-2,{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{7,-17,17,-6},{4,6,11,26},20]

gf=x FindGeneratingFunction[seq,x]

Series[gf,{x,0,10}]

CoefficientList[Series[((4-22 x+37 (x^2)-17 (x^3))/((((-1+x))^2) (1-5 x+6 (x^2)))),{x,0,20}],x]

KirchhoffIndex

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

FindSequenceFunction[seq,n]//Factor

KirchhoffSumIndex

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

FindSequenceFunction[seq,n]//Factor

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Book",n},"LaplacianMatrix"],{n,20}]

LaplacianPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"LaplacianPolynomial"],{n,k0=1,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]

LeafCount

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

LongestCycleCount

Same as |HC|

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

LovaszNumber

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

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"MatchingGeneratingPolynomial"],{n,k0=1,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]

MatchingNumber

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

FindSequenceFunction[seq,n]

MatchingPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"MatchingPolynomial"],{n,k0=1,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]

MaximalCliqueCount

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

FindSequenceFunction[seq,n]

MaximalIndependentEdgeSetCount

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

FindSequenceFunction[seq,n]

MaximalIndependentVertexSetCount

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

FindSequenceFunction[seq,n]

MaximalIrredundantSetCount

A291064

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

FindSequenceFunction[DeleteMissing@seq,n]

FindLinearRecurrence[DeleteMissing@seq]

FindLinearRecurrence[Rest@DeleteMissing@seq]

seq=LinearRecurrence[{6,-13,12,-4},{4,9,26,71,180},40]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((4-15 x+24 (x^2)-16 (x^3))/(((1-3 x+2 (x^2)))^2)),{x,0,20}],x]

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

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

MaximumLeafNumber

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

MaximumVertexDegree

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

MeanDistance

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

FindSequenceFunction[seq,n]

MinimalDominatingSetCount

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

FindSequenceFunction[DeleteMissing@seq,n]//Factor

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]//Factor

MinimalDominatingSetSignature

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

FindSequenceFunction[{6,14,30,62,126},n-2]//FullSimplify

MinimalEdgeCoverCount

A290699

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

Table[GraphData[{"Book",n},"EdgeCoverCount"],{n,20}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]//Factor

seq=Table[(2^n)-n+(n^2),{n,40}]

FindLinearRecurrence[seq]

LinearRecurrence[{5,-9,7,-2},{2,6,14,28},20]

gf=x FindGeneratingFunction[seq,x]

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

MinimumCliqueCoveringCount

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

MinimumConnectedDominatingSetCount

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

MinimumDistinguishingLabelingCount

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

MinimumDominatingSetCount

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

MinimumEdgeCoverCount

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

MinimumVertexDegree

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

MolecularTopologicalIndex

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]//Factor

Nonhamiltonian

Table[GraphData[{"Book",n},"Nonhamiltonian"],{n,20}]

NormalizedLaplacianMatrix

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

OddChordlessCycleCount

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

PathCount

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

PathPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

p=.

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[{"Book",n},"PentagonCount"],{n,20}]

Perfect

Table[GraphData[{"Book",n},"Perfect"],{n,20}]

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"RankPolynomial"],{n,k0=1,kmax=20}]/.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]

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

FullSimplify[poly2-poly]

RectilinearCrossingNumber

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

ReliabilityPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"ReliabilityPolynomial"],{n,k0=1,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[{"Book",n},"SigmaPolynomial"],{n,k0=1,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

A006234

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[Take[seq,7],x]

SquareCount

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

StabilityIndex

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

FindSequenceFunction[seq,n]

TopologicalIndex

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

FindSequenceFunction[seq,n]

ToroidalCrossingNumber

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

Traceable

Table[GraphData[{"Book",n},"Traceable"],{n,20}]

TriangleCount

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

TuttePolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"Book",n},"Untraceable"],{n,20}]

VertexConnectivity

Table[GraphData[{"Book",n},"VertexConnectivity"],{n,20}]

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

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

FindSequenceFunction[seq,n]

VertexCoverPolynomial

(poly=Factor[Table[GraphData[{"Book",n},"VertexCoverPolynomial"],{n,k0=1,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[{"Book",n},"VertexDegrees"],{n,20}]

VertexTransitive

Table[GraphData[{"Book",n},"VertexTransitive"],{n,20}]

WienerIndex

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

FindSequenceFunction[seq,n]

WienerSumIndex

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

FindSequenceFunction[seq,n]//Factor

GraphData

GraphClassString

GraphClassString[GraphData["Book"]]

GraphDataString

Do[Print[GraphDataString[BookGraph[m],
"Name"->ToString[m]<>"-book graph",
"StandardName"->{"Book",m},
"Classes"->{"Book"},
"NotationRules"->{"Book"->m},
"Information"->"BookGraph",
TimeConstraint->120
]],{m,3,20}]