Web Graph

Author

Eric W. Weisstein
August 9, 2017

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

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

Graphic

Graphs.m

GraphicsRow[Show[WebGraph[#,VertexSize->{"Scaled",.07},System`VertexStyle->Red,System`EdgeStyle->Black],PlotLabel->Style[Subscript[W,#],FontFamily->"Times"]]&/@Range[3,8],ImageSize->500,Dividers->All]

Properties

Acyclic

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

AdjacencyMatrixCount

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

FindSequenceFunction[seq,n-2]//FullSimplify

Arboricity

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

ArcTransitive

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

ArcTransitivity

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

AutomorphismCount

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

BalabanIndex

Table[GraphData[{"Web",n},"BalabanIndex"],{n,3,5}]

Bandwidth

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

BridgeCount

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

BurningNumber

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

CharacteristicPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

ChordlessCycleCount

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

FindSequenceFunction[seq,n-2]

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

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

Table[Piecewise[{{3,n==3}},LucasL[n]+n+2Cos[n π/3]],{n,3,20}]

ChromaticInvariant

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

ChromaticNumber

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

ChromaticPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Circumference

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

FindSequenceFunction[seq,n-2]

Classes

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

CliqueCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

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

CliqueCoveringNumber

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

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

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

CliqueNumber

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

CliquePolynomial

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

FindLinearRecurrence[Rest@poly]

poly2=Table[Piecewise[{{1+9 x+12 (x^2)+2 (x^3),n==3}},1+3n x+4n (x^2)],{n,k0,kmax}]

FullSimplify[poly2-poly]

ComplementOddChordlessCycleCount

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

ConnectedComponentCount

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

ConnectedDominatingSetCount

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

FindSequenceFunction[DeleteCases[seq,_Missing],n-2]

ConnectedDominationNumber

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

FindSequenceFunction[DeleteCases[seq,_Missing],n-2]

ConnectedDominationPolynomial

(poly=DeleteCases[Factor@Table[GraphData[{"Web",n},"ConnectedDominationPolynomial"][x],{n,k0=3,20}],_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[XXX,{n,3,20}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

A286187

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

seq={6,33,188,985,4990,24645,119712,574225,2727218,12847821,60115060,279652793,1294441894,5965567125,27387631368,125308264225,571591760602,2600204421405,11799376912220,53424388364873,241398575303374,1088727972172389,4901842528232304,22034981672761649};

FindLinearRecurrence[DeleteCases[seq,_Missing]]

Take[seq,8]

LinearRecurrence[{12,-50,80,-17,-76,52,16,-16},{6,33,188,985,4990,24645,119712,574225},20]

LinearRecurrence[{12,-50,80,-17,-76,52,16,-16},{6,33,188,985,4990,24645,119712,574225},20]

FindSequenceFunction[seq,n-2]

gf=x FindGeneratingFunction[DeleteCases[seq,_Missing],x]

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

Table[RootSum[4+2 #1-5 ((#1)^2)+((#1)^3)&,#^n&]-(1/1537150592)n (768575296+
Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,1] ((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,3])^n)+Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,2]((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,2])^n)+ Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,3] ((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,1])^n) ),{n,20}]//FullSimplify

FullSimplify[(1/1537150592){Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,1] ,Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,2],Root[1609829234487607221847851008+113498477586661376 #1-2305725888 ((#1)^2)+((#1)^3)&,3] }]

Table[((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,1])^n)+((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,2])^n)+((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,3])^n)-n ((1/2)+
Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,1]((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,3])^n)+
Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,2]((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,2])^n)+ 
Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,3]((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,1])^n) ),{n,20}]//FullSimplify

Table[((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,1])^n)(1-n  Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,3])+
((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,2])^n)(1-n Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,2])+
((Root[4+2 #1-5 ((#1)^2)+((#1)^3)&,3])^n)(1-n  Root[203+22 #1-687 ((#1)^2)+458 ((#1)^3)&,1])-
(1/2)n,{n,20}]//FullSimplify

On 05/27/2017 05:55 AM, Eric Weisstein wrote:
> Hi, Adam.
>
> Any idea if the root products in:
>
> RootSum[4 + 2 #1 - 5 #1^2 + #1^3 &, #^n &] - n (1/2 +
>      Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 1] Root[
>        4 + 2 #1 - 5 #1^2 + #1^3 &, 3]^n +
>      Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 2] Root[
>        4 + 2 #1 - 5 #1^2 + #1^3 &, 2]^n +
>      Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 3] Root[
>        4 + 2 #1 - 5 #1^2 + #1^3 &, 1]^n )
>
> (https://oeis.org/A286187)
>
> can somehow themselves be written as a root sum?
>
> Many thanks,
> -E
>

Yes, the first Root in each product belongs to the field generated by the second Root, with the same coefficientsfor all products.

In[1]:= expr=RootSum[4 + 2 #1 - 5 #1^2 + #1^3 &, #^n &] - n (1/2 +
            Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 1] Root[
              4 + 2 #1 - 5 #1^2 + #1^3 &, 3]^n +
            Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 2] Root[
              4 + 2 #1 - 5 #1^2 + #1^3 &, 2]^n +
            Root[203 + 22 #1 - 687 #1^2 + 458 #1^3 &, 3] Root[
              4 + 2 #1 - 5 #1^2 + #1^3 &, 1]^n );

In[2]:= f1=4 + 2 #1 - 5 #1^2 + #1^3 &;

In[3]:= f2=203 + 22 #1 - 687 #1^2 + 458 #1^3 &;

In[4]:= an=ToNumberField[Root[f2, #], Root[f1, 4-#]]&/@Range[3];

In[5]:= coeffs=Last/@an;

In[6]:= SameQ@@coeffs
Out[6]= True

In[7]:= g=#^n-n (coeffs[[1]].{1, #, #^2}) #^n;

In[8]:= (ee=RootSum[f1, Evaluate[g]&]-n/2)//InputForm
Out[8]//InputForm=
-n/2 + RootSum[4 + 2*#1 - 5*#1^2 + #1^3 & ,
   458*#1^n - 494*n*#1^n + 12*n*#1^(1 + n) + 35*n*#1^(2 + n) & ]/458

In[9]:= Table[RootReduce[expr-ee], {n, 7}]
Out[9]= {0, 0, 0, 0, 0, 0, 0}

Table[RootSum[4+2#-5#^2+#^3&,458#^n-494n#^n+12n#^(n+1)+35n#^(n+2)&]/458-n/2,{n,20}]

ConnectedInducedSubgraphPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

poly2=Table[XXX,{n,3,20}]

FullSimplify[poly2-poly]

CrossingNumber

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

CycleCount

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

FindSequenceFunction[seq,n-2]//FullSimplify

CyclePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Cyclic

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

CyclomaticNumber

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

FindSequenceFunction[seq,n-2]

DetourIndex

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

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

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

DetourMatrix

MatrixForm/@Table[GraphData[{"Web",n},"DetourMatrix"],{n,3,5}]

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

DetourPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Diameter

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

DistanceMatrix

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

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

DistancePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

DistinguishingNumber

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

DomaticNumber

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

DominatingSetCount

A287474

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]//FullSimplify

FullSimplify[Root[-81-81 #1-31 ((#1)^2)+((#1)^3)&,1] ((Root[-9-3 #1-5 ((#1)^2)+((#1)^3)&,1])^(-2))]

LinearRecurrence[{5,3,9},{5,31,197},20]

Table[RootSum[-9-3 #-5 ((#)^2)+((#)^3)&,#^n&],{n,20}]

DominationNumber

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

DominationPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EdgeChromaticNumber

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

EdgeConnectivity

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

EdgeCount

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

FindSequenceFunction[seq,n-2]

EdgeCoverCount

A287082

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

FindLinearRecurrence[seq]

seq=LinearRecurrence[{6,8},{6,52},20]

FindSequenceFunction[seq,n-2]//FullSimplify

Table[(((3-(Sqrt[17])))^n)+(((3+(Sqrt[17])))^n),{n,20}]//Expand

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

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

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

EdgeCoverNumber

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

FindSequenceFunction[seq,n-2]//FullSimplify

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

EdgeCoverPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

EulerianCycleCount

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

FaceCount

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

FlowPolynomial

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

FractionalChromaticNumber

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

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

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n≥3]

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

FractionalEdgeChromaticNumber

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

Girth

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

Gonality

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

HamiltonConnected

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

HamiltonDecompositionCount

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

Hamiltonian

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

HamiltonianCycleCount

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

HamiltonianNumber

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

HamiltonianPathCount

seq=Table[GraphData[{"Web",n},"HamiltonianPathCount"],{n,3,20}]

HamiltonianWalkCount

seq=Table[GraphData[{"Web",n},"HamiltonianWalkCount"],{n,3,20}]

HararyIndex

seq=Table[GraphData[{"Web",n},"HararyIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

HexagonCount

seq=Table[GraphData[{"Web",n},"HexagonCount"],{n,3,20}]

IdiosyncraticPolynomial

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

coef=Factor/@FindLinearRecurrence[Drop[poly,-4]]

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[{"Web",n},"IndependenceNumber"],{n,3,20}]

FindSequenceFunction[seq,n-2]//FullSimplify

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

IndependencePolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

(poly2=Factor@Table[GraphData[{"Web",n},"IndependencePolynomial"][x],{n,k0=3,kmax=20}]/.f_Function:>f[x])//Column

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]

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

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"Web",n},"IndependenceRatio"],{n,3,20}]

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

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

IndependentEdgeSetCount

A192857

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

Table[RootSum[1-#-7 #^2-3 #^3+#^4&,#^n&],{n,25}]

RootSum[1-#-7 #^2-3 #^3+#^4&,#^Range[0,25]&]

FindLinearRecurrence[DeleteCases[seq,_Missing]]

seq=LinearRecurrence[{3,7,1,-1},{4,3,23,93},20]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((4-9 x-14 (x^2)-(x^3))/(1-3 x-7 (x^2)-(x^3)+(x^4))),{x,0,20}],x]

IndependentVertexSetCount

A287485

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

FindLinearRecurrence[seq]

LinearRecurrence[{2,8,4},{68,304,1232},{-1,1}]

seq={2,20,68,304,1232,5168,21408,89088,370112,1538560,6394368,26577664,110464512,459127808,1908282368,7931445248,32965660672,137016012800,569483091968,2366956929024};

FindSequenceFunction[seq,n]

Table[RootSum[-4-8 #-2 #^2+#^3&,#^n&],{n,20}]

LinearRecurrence[{2,8,4},{2,20,68},20]

IntersectionNumber

seq=Table[GraphData[{"Web",n},"IntersectionNumber"],{n,3,20}]

IrredundantSetCount

A290591

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

seq={4,22,112,618,3044,14026,70340,347074,1713820,8430362,41516292,204628626,1008216460,4967464650,24473678612,120581823170,594106884060,2927147525626,14421969381220,71056635221618,350094259434668,1724905262555178,8498562511858612};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{4,3,4,22,12,-164,80,-24,80},{4,22,112,618,3044,14026,70340,347074,1713820},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[-((2 (2+3 x+6 (x^2)+44 (x^3)+30 (x^4)-492 (x^5)+280 (x^6)-96 (x^7)+360 (x^8)))/(-1+4 x+3 (x^2)+4 (x^3)+22 (x^4)+12 (x^5)-164 (x^6)+80 (x^7)-24 (x^8)+80 (x^9))),{x,0,20}],x]

res=Refine[SeriesCoefficient[gf,{x,0,n}],n>2]

terms=List@@res;

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

terms[[All,1]]

Table[RootSum[-80+24 #-80 ((#)^2)+164 ((#)^3)-12 ((#)^4)-22 ((#)^5)-4 ((#)^6)-3 ((#)^7)-4 ((#)^8)+((#)^9)&,((#)^n)&],{n,20}]

KirchhoffIndex

seq=Table[GraphData[{"Web",n},"KirchhoffIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

KirchhoffSumIndex

seq=Table[GraphData[{"Web",n},"KirchhoffSumIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Web",n},"LaplacianMatrix"],{n,3,20}]

LaplacianPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"Web",n},"LeafCount"],{n,3,20}]

LongestCycleCount

seq=Table[GraphData[{"Web",n},"LongestCycleCount"],{n,3,20}]

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

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

LovaszNumber

seq=Table[GraphData[{"Web",n},"LovaszNumber"],{n,3,20}]

MatchingGeneratingPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

MatchingNumber

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

FindSequenceFunction[seq,n-2]

FullSimplify[%]

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

MatchingPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

MaximalCliqueCount

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

FindSequenceFunction[seq,n-2]

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

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

MaximalIndependentEdgeSetCount

A287497

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

FindLinearRecurrence[DeleteMissing[seq]]

LinearRecurrence[{1,4,3,-2,0,1},{22,53,166,432,1212,3341},{-1,2}]

seq=LinearRecurrence[{1,4,3,-2,0,1},{22,53,166,432,1212,3341},{-1,20}]

FindSequenceFunction[DeleteMissing[seq],n-2]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

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

N[(List@@soln)/.n->0]//Chop

Rest@CoefficientList[Series[((x (-1-8 x-9 (x^2)+8 (x^3)-6 (x^5)))/(-1+x+4 (x^2)+3 (x^3)-2 (x^4)+(x^6))),{x,0,20}],x]

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

LinearRecurrence[{1,4,3,-2,0,1},{1,9,22,53,166,432},30]

MaximalIndependentVertexSetCount

A287498

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

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n-2]

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

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

Table[RootSum[-1-3 #-2 ((#)^2)+((#)^4)&,#^n&],{n,30}]

LinearRecurrence[{0,2,3,1},{0,4,9,12},20]

MaximumCliqueCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

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

MaximumIndependentEdgeSetCount

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

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

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

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

MaximumIndependentVertexSetCount

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]//FullSimplify

FullSimplify[%136,Mod[n,2]==#]&/@{0,1}

MaximumLeafNumber

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

FindSequenceFunction[seq,n-2]

MaximumVertexDegree

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

MeanDistance

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

FindSequenceFunction[seq,n-2]//FullSimplify

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

MinimalDominatingSetCount

A290381

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]

seq=Import["https://oeis.org/A290381/b290381.txt","Table"][[All,2]]

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

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

terms=Simplify[SeriesCoefficient[gf,{x,0,n}],n>2]

N[List@@terms/.n->0]//Chop

terms[[All,1]]

Table[RootSum[8-4 #1-4 ((#1)^2)-5 ((#1)^3)-2 ((#1)^4)-((#1)^5)+((#1)^6)&,#^n&],{n,3,20}]

MinimalDominatingSetSignature

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

MinimalEdgeCoverCount

A001644

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

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

MinimalEdgeCoverSignature

seq=Table[GraphData[{"Web",n},"MinimalEdgeCoverSignature"],{n,3,20}]//Column

MinimumCliqueCoveringCount

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

MinimumConnectedDominatingSetCount

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

FindSequenceFunction[DeleteCases[seq,_Missing],n]

MinimumDistinguishingLabelingCount

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

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq]]

MinimumDominatingSetCount

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

MinimumEdgeCoverCount

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

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

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

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

MinimumVertexDegree

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

MolecularTopologicalIndex

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

FindSequenceFunction[seq,n-2]//FullSimplify

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

Nonhamiltonian

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

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"Web",n},"NormalizedLaplacianMatrix"],{n,3,20}]

OddChordlessCycleCount

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

FindGeneratingFunction[seq[[3;;All;;2]],x]

PathCount

A288037

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]//Factor

Table[(1/4) n (-104+89 (2^n)-52 n-8 (n^2)),{n,3,20}]

LinearRecurrence[{8,-26,44,-41,20,-4},{285,984,2855,7488,18431,43504},18]

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

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

gf=(((x^3) (285-1296 x+2393 (x^2)-2308 (x^3)+1146 (x^4)-232 (x^5)))/((((1-x))^4) (((1-2 x))^2)));

CoefficientList[Series[((285-1296 x+2393 (x^2)-2308 (x^3)+1146 (x^4)-232 (x^5))/((((1-x))^4) (((1-2 x))^2))),{x,0,20}],x]

PathPolynomial

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

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

PentagonCount

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

Perfect

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

ProjectivePlaneCrossingNumber

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

Radius

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

RankPolynomial

(poly=Factor@Table[GraphData[{"Web",n},"RankPolynomial"],{n,k0=3,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]

RectilinearCrossingNumber

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

ReliabilityPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SigmaPolynomial

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

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

SpanningTreeCount

A006235

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

FindSequenceFunction[seq,n-2]//FullSimplify

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

Table[n(LucasL[2n,Sqrt[2]]/2-1),{n,10}]//N//Round

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

Table[LucasL[2n,Sqrt[2]]/2-ChebyshevT[n,2],{n,5}]//FullSimplify

N[%]//Chop

ChebyshevT[Range[0,20],2]

SquareCount

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

StabilityIndex

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

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-2]

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

TopologicalIndex

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

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

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

Traceable

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

TriangleCount

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

TuttePolynomial

(poly=Factor@Table[GraphData[{"Web",n},"TuttePolynomial"][x,y],{n,k0=3,kmax=12}/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

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

TeXForm[%]

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

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

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

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

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

FullSimplify[poly2-poly]

Untraceable

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

VertexConnectivity

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

VertexCount

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

FindSequenceFunction[seq,n]

VertexCoverNumber

Related to but not the same as IndependenceNumber

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

FindSequenceFunction[seq,n]//FullSimplify

gf=x FindGeneratingFunction[seq,x]

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

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

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"Web",n},"VertexCoverPolynomial"][x],{n,k0=3,kmax=12}/.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[{"Web",n},"VertexDegrees"],{n,3,20}]

VertexTransitive

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

WienerIndex

A180576

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

FindSequenceFunction[seq,n-2]//FullSimplify

FullSimplify[Expand[(1/16) (((-1))^n) n (9+(((-1))^n) (-25+2 n (40+9 n)))]/.(((-1))^(2n)):>1]

Table[n (-25+9 (((-1))^n)+2 n (40+9 n))/16,{n,20}]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length@coef]

LinearRecurrence[{2,1,-4,1,2,-1},{69,148,255,417,616,888},{-1,1}]

seq=LinearRecurrence[{2,1,-4,1,2,-1},{4,27,69,148,255,4178},20]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((4+19 x+11 (x^2)-(x^3)-6 (x^4)+3761 (x^5))/((((-1+x))^4) (((1+x))^2))),{x,0,20}],x]

WienerSumIndex

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

FindSequenceFunction[seq,n]

GraphData

Do[Print[GraphDataString[WebGraph[n],
"Name"->ToString[n]<>"-web graph",
"StandardName"->{"Web",n},
"Classes"->{"Web"},
"NotationRules"->{"Web"->n},
"Skip"->{"Perfect"},
TimeConstraint->120
]],{n,3,20}]