Minimal Edge Cover

Author

Eric W. Weisstein
February 22, 2022

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

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

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

Table

GraphDataCountTable["MinimalEdgeCoverCount",Join[{"Andrasfai"->290702,"Antiprism"->290698,"Barbell"->290715,"Book"->290699,"CocktailParty"->351589,"Complete"->053530,"CompleteBipartite"->210655,"CrossedPrism"->351586,"Crown"->290755,"Cycle"->001608,"CycleComplement"->351587,"DiagonalIntersection"->351590,"Dipyramid"->297713,"Empty"->000007,"FibonacciCube"->297118,"Gear"->290762,{"Grid",2}->288027,"Helm"->000027,"Ladder"->288029,"MoebiusLadder"->290470,"Pan"->000931,"Path"->000931,"PathComplement"->351588,"Prism"->290700,"Star"->057427,"Sun"->290763,"TriangularHoneycombAcuteKnight"->351591,"TriangularSnake"->308589,"Web"->001644,"Wheel"->290471},Thread[{"Alkane","Centipede","LadderRung","Sunlet"}->000012]]]

Parametrized families

Andrasfai

A290702

seq=Table[GraphData[{"Andrasfai",n},"MinimalEdgeCoverCount"],{n,10}]

Table[GraphData[{"Andrasfai",n},"EdgeCoverCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

Antiprism

Entity["Graph",{"Antiprism",n}]["MinimalEdgeCoverCount"]

A290698

seq=Table[GraphData[{"Antiprism",n},"MinimalEdgeCoverCount"],{n,3,25}]

Table[GraphData[{"Antiprism",n},"EdgeCoverCount"],{n,3,20}]

seq={2,14,74,286,1157,4778,19623,80478,330293,1355629,5563527,22832914,93707772,384582275,1578347684,6477630782,26584574434,109104640685,447771795953,1837681518261,7541951930181,30952609765223,127031312347552,521343900861138};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{2,5,12,11,5,-3,-4,-1,1},{2,14,74,286,1157,4778,19623,80478,330293},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((-2-10 x-36 (x^2)-44 (x^3)-25 (x^4)+18 (x^5)+28 (x^6)+8 (x^7)-9 (x^8))/(-1+2 x+5 (x^2)+12 (x^3)+11 (x^4)+5 (x^5)-3 (x^6)-4 (x^7)-(x^8)+(x^9))),{x,0,20}],x]

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

part1=FullSimplify[(((((-1))^((2 (2+n))/3)) (-2+(((-1))^(1/3)))+(2+(((-1))^(2/3))) (E^(-(2/3) I n π)))/(1+(((-1))^(1/3)))),n∈Integers&&n>0]

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((#1)^6)+((#1)^7)&,2]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,3]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,4]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,5]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]) (Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]))+(((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^n) (2+6 Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]-6 ((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^2)-24 ((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^3)-15 ((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^4)-18 ((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^5)+7 ((Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])^6)))/((-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,1]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,2]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,3]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,4]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,5]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7]) (-Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,6]+Root[-1+2 #1+3 ((#1)^2)-2 ((#1)^3)-6 ((#1)^4)-3 ((#1)^5)-3 ((#1)^6)+((#1)^7)&,7])));

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

terms[[All,1]]

Table[2 Cos[((2 n π)/3)]+RootSum[-1+2 #+3 ((#)^2)-2 ((#)^3)-6 ((#)^4)-3 ((#)^5)-3 ((#)^6)+((#)^7)&,#^n&],{n,20}]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"MinimalEdgeCoverCount"],{n,6}]

Barbell

A290715

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

seq=Import["https://oeis.org/A290715/b290715.txt","Table"][[All,2]]

FindLinearRecurrence[seq]//Head//Timing

FindSequenceFunction[DeleteMissing@seq,n-2]//Head//Timing

gf=(x^3)FindGeneratingFunction[DeleteMissing@seq,x2]//Head//Timing

b[n_]:=n!Sum[(1/(k!))(Binomial[k,n-k](2^(k-n))(((-1))^k)+Sum[Binomial[k,j]Sum[((j^(i-j))/((i-j)!))Binomial[k-j,n-i-k+j](2^(i-j+k-n))(((-1))^(k-j)),{i,j,n-k+j}],{j,k}]),{k,n}]

Table[((b[n])^2)+b[n-1](b[n-1]+2+2Sum[Binomial[n-1,i]b[i],{i,n-2}]),{n,20}]

Bishop

seq=Table[GraphData[{"Bishop",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

Book

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]

Centipede

seq=Table[GraphData[{"Centipede",n},"MinimalEdgeCoverCount"],{n,10}]

CocktailParty

A351589

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

Do[Print[{k,FindLinearRecurrence[DeleteMissing@Drop[seq,k]]//Head//Timing}],{k,4}]

Do[Print[{k,FindSequenceFunction[DeleteMissing@Drop[seq,k],n]//Head//Timing}],{k,4}]

Do[Print[{k,FindGeneratingFunction[DeleteMissing@Drop[seq,k],x]//Head//Timing}],{k,4}]

(PARI) a(n)={my(x=x+O(x^(2*n+1)), p=exp(-x - x^2/2 + x*exp(x)), q=2*exp(x) - 1); sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k)!*polcoef(q^(n-k)*p, 2*k))}

Table[With[{p=Series[Exp[-x-x^2/2+x Exp[x]],{x,0,2n+1}],q=Series[2 Exp[x]-1,{x,0,2n+1}]},
Sum[(((-1))^(n-k))Binomial[n,k](2k)!Coefficient[(q^(n-k))p,x,2k],{k,0,n}]],{n,10}]

Series[Exp[-x-x^2/2+x Exp[x]],{x,0,10}]

CoefficientList[%,x]

Transpose[NumeratorDenominator[%]]

SeriesCoefficient[Exp[-x-x^2/2+x Exp[x]],{x,0,n}]

Series[2 Exp[x]-1,{x,0,5}]

FullSimplify[SeriesCoefficient[2 Exp[x]-1,{x,0,2k}],k>=0]

Table[(2/((2k)!)),{k,0,3}]

Series[Exp[-x-x^2/2+x Exp[x]](2 Exp[x]-1),{x,0,10}]

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

SeriesCoefficient[Exp[-x-x^2/2+x Exp[x]](2 Exp[x]-1),{x,0,10}]

Table[With[{p=Series[Exp[-x-x^2/2+x Exp[x]],{x,0,2n+1}],q=Series[2 Exp[x]-1,{x,0,2n+1}]},
Sum[(((-1))^(n-k))Binomial[n,k](2k)!Coefficient[(q^(n-k))p,x,2k],{k,0,n}]],{n,10}]

Table[Sum[(((-1))^(n-k))Binomial[n,k](2k)!SeriesCoefficient[(((Exp[-x-x^2/2+x Exp[x]]))^(n-k))(2 Exp[x]-1),{x,0,2k}],{k,0,n}],{n,10}]

Complete

Entity["Graph",{"Complete",n}]["MinimalEdgeCoverCount"]

n! (InactiveSum[(1/(\[FormalK]!))((((-1))^\[FormalK]) (2^(\[FormalK]-n)) (Binomial[\[FormalK], n-\[FormalK]])+(InactiveSum[(Binomial[\[FormalK], \[FormalJ]]) (InactiveSum[(((((-1))^(\[FormalK]-\[FormalJ])) (2^(\[FormalI]-\[FormalJ]+\[FormalK]-n)) (\[FormalJ]^(\[FormalI]-\[FormalJ])))/((\[FormalI]-\[FormalJ])!)) (Binomial[\[FormalK]-\[FormalJ], -\[FormalI]+\[FormalJ]-\[FormalK]+n]), \[FormalI], \[FormalJ], \[FormalJ]-\[FormalK]+n]), \[FormalJ], \[FormalK]]), \[FormalK], n]))

A053530

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

Do[Print[{n,Length@MinimalSets[FindAllEdgeCovers[System`CompleteGraph[n]]]//Timing}],{n,2,7}]

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

seq=With[{n=40},CoefficientList[Series[Exp[-x-(x^2)/2+x Exp[x]],{x,0,n}],x]Range[0,n]!]

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

FindLinearRecurrence[seq]

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

Table[n!Sum[(1/(k!))(Binomial[k,n-k](2^(k-n))(((-1))^k)+Sum[Binomial[k,j]Sum[((j^(i-j))/((i-j)!))Binomial[k-j,n-i-k+j](2^(i-j+k-n))(((-1))^(k-j)),{i,j,n-k+j}],{j,k}]),{k,n}],{n,40}]

Sum[j^(i-j)/(i-j)!Binomial[k-j,n-i-k+j](2^(i-j+k-n))(((-1))^(k-j)),{i,j,n-k+j}]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["MinimalEdgeCoverCount"]

ser=Series[Exp[x Exp[y]+y Exp[x]-x-y-x y]-1,{x,0,20},{y,0,20}];

Block[{m,n},Sort[{{m,n}="CompleteBipartite"/.GraphData[#,"NotationRules"],GraphData[#,"MinimalEdgeCoverCount"],m!n!Coefficient[ser,x^m y^n]}&/@GraphData["CompleteBipartite"]]]

DeleteCases[%,{_,x_,x_}]

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["MinimalEdgeCoverCount"]

A210655

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

T[p_,q_]:=T[p,q]=If[p==1||q===1,1,Sum[Binomial[q,r]*T[p-1,q-r],{r,2,q-1}]+q*Sum[Binomial[p-1,s]*T[p-s-1,q-1],{s,0,p-2}]];a[n_]:=T[n,n];Table[a[n],{n,1,20}]

With[{ser=Series[Exp[x Exp[y]+y Exp[x]-x-y-x y]-1,{x,0,20},{y,0,20}]},Table[(((n!))^2)Coefficient[ser,x^n y^n],{n,20}]]

SeriesCoefficient[Exp[x Exp[y]+y Exp[x]-x-y-x y]-1,{x,0,n},{y,0,n}]

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["MinimalEdgeCoverCount"]

Block[{k,m,n},Sort[{{k,m,n}="CompleteTripartite"/.GraphData[#,"NotationRules"],GraphData[#,"MinimalEdgeCoverCount"],(k^2)+(m^2)+(n^2)}&/@GraphData["CompleteTripartite"]]]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"MinimalEdgeCoverCount"],{n,6}]

Table[Piecewise[{{3,n==1}},3n^2+3],{n,10}]

CrossedPrism

A351586

seq=Table[GraphData[{"CrossedPrism",2n},"MinimalEdgeCoverCount"],{n,2,10}]

seq=LinearRecurrence[{5,12,40,56,-8,-32},{5,49,425,3137,22985,172081},20]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

bug(420571): SeriesCoefficient[(x (5 + 24 x + 120 x^2 + 224 x^3 - 40 x^4 - 192 x^5))/(1 - 5 x - 12 x^2 - 40 x^3 - 56 x^4 + 8 x^5 + 32 x^6), {x, 0, n}] could give simple RootSum instead of complicated expression involving hidden unities

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

Table[expr,{n,10}]//Round

expr//N

res=Cases[expr,c_^n b_:>{c^n,ToAlgebraicRoot[b]}]

Total[Times@@@res]

Table[((NumericalApproximation[Root, (RowDefault["0.649", FEExpressionsNumericalApproximationElider]), Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,1], ])^n)+((NumericalApproximation[Root, (RowDefault["7.46", FEExpressionsNumericalApproximationElider]), Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,2], ])^n)+((NumericalApproximation[Root, (RowDefault["-1.13", FEExpressionsNumericalApproximationElider])-(RowDefault["0.104", FEExpressionsNumericalApproximationElider]) I, Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,3], ])^n)+((NumericalApproximation[Root, (RowDefault["-1.13", FEExpressionsNumericalApproximationElider])+(RowDefault["0.104", FEExpressionsNumericalApproximationElider]) I, Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,4], ])^n)+((NumericalApproximation[Root, (RowDefault["-0.426", FEExpressionsNumericalApproximationElider])-(RowDefault["2.23", FEExpressionsNumericalApproximationElider]) I, Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,5], ])^n)+((NumericalApproximation[Root, (RowDefault["-0.426", FEExpressionsNumericalApproximationElider])+(RowDefault["2.23", FEExpressionsNumericalApproximationElider]) I, Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,6], ])^n),{n,5}]//N

(NumericalApproximation[Root, (RowDefault["0.649", FEExpressionsNumericalApproximationElider]), Root[32+8 #1-56 ((#1)^2)-40 ((#1)^3)-12 ((#1)^4)-5 ((#1)^5)+((#1)^6)&,1], ])//InputForm

Table[RootSum[32+8#-56#^2-40#^3-12#^4-5#^5+#^6&,#^n&],{n,20}]

CoefficientList[Series[((5+24 x+120 (x^2)+224 (x^3)-40 (x^4)-192 (x^5))/(1-5 x-12 (x^2)-40 (x^3)-56 (x^4)+8 (x^5)+32 (x^6))),{x,0,20}],x]

Crown

A290755

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

Do[Print[{k,FindLinearRecurrence[DeleteMissing@Drop[seq,k]]//Head//Timing}],{k,0,3}]

Do[Print[{k,FindGeneratingFunction[DeleteMissing@Drop[seq,k],x]//Head//Timing}],{k,0,3}]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"MinimalEdgeCoverCount"],{n,3,8}]

Cycle

Perrin numbers

Entity["Graph",{"Cycle",n}]["MinimalEdgeCoverCount"]

A001608

== MaximalIndependentEdgeSetCount == MaximalIndependentVertexSetCount == MinimalVertexCoverCount

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

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

Table[GraphData[{"Cycle",n},"MaximalIndependentEdgeSetCount"],{n,3,20}]

Table[GraphData[{"Cycle",n},"MaximalIndependentVertexSetCount"],{n,3,20}]

Table[GraphData[{"Cycle",n},"MinimalVertexCoverCount"],{n,3,20}]

DiagonalIntersection

A351590

seq=Table[GraphData[{"DiagonalIntersection",n},"MinimalEdgeCoverCount"],{n,3,14}]

Empty

Entity["Graph",{"Empty",n}]["MinimalEdgeCoverCount"]

FindEdgeCover[GraphData[{"Empty",1}]]

GraphData[{"Empty",1},"MinimalEdgeCoverCount"]

GraphData["P2+K1","EdgeCovers"]

GraphData[{"Empty",1},"EdgeCovers"]

GraphData[{"Empty",2},"EdgeCovers"]

GraphData[{"Empty",2},"EdgeCoverCount"]

seq=Table[GraphData[{"Empty",n},"MinimalEdgeCoverCount"],{n,30}]

FindEdgeCover[System`Graph[{1},{}]]

FindEdgeCover[System`Graph[{1,2},{}]]

FibonacciCube

A297118

seq=Table[GraphData[{"FibonacciCube",n},"MinimalEdgeCoverCount"],{n,15}]

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"MinimalEdgeCoverCount"],{n,10}]

Gear

A290762

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

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

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

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

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

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

FullSimplify[RootReduce[res]]

Expand[%81]//RootReduce

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

GeneralizedPetersen

Sort[{"GeneralizedPetersen"/.GraphData[#,"NotationRules"],GraphData[#,"MinimalEdgeCoverCount"]}&/@GraphData["GeneralizedPetersen"]]

Grid 3×n

Entity["Graph",{"Grid",{m,n}}]["MinimalEdgeCoverCount"]

A288030

seq=Table[GraphData[{"Grid",Sort[{3,n}]},"MinimalEdgeCoverCount"],{n,10}]

seq=Import["https://oeis.org/A288030/b288030.txt","Table"][[All,2]];

coef=FindLinearRecurrence[seq]

Length[%]

Take[seq,Length[coef]]

gf=x FindGeneratingFunction[seq,x]

Series[gf=x*(1+3*x+14*x^2+5*x^3+20*x^4-12*x^5+15*x^6+6*x^7+2*x^8-82*x^9-5*x^10-3*x^11+29*x^12+8*x^14-2*x^15-2*x^16-x^17)/(1-3*x-6*x^2-35*x^3+7*x^4-15*x^5+55*x^6-31*x^7-36*x^8-36*x^9+124*x^10+8*x^12-32*x^13-x^14-9*x^15+2*x^16+2*x^17+x^18),{x,0,20}]

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

Grid m×n

Entity["Graph",{"Grid",{m,n}}]["MinimalEdgeCoverCount"]

A288025

Block[{m,n},Sort[{{m,n}="Grid"/.GraphData[#,"NotationRules"],GraphData[#,"MinimalEdgeCoverCount"],Piecewise[{{2,Min[m,n]==1}},m n+2]}&/@Select[GraphData["Grid"],Length["Grid"/.GraphData[#,"NotationRules"]]==2&]]]

data=Import["https://oeis.org/A288025/b288025.txt","Table"][[All,2]];

Table[data[[Binomial[i,2]+(1/2) (j-1) (2 i+j)+1]],{i,10},{j,If[i==10,8,i]}]//Column

Table[GraphData[{"Grid",{m,n}},"MinimalEdgeCoverCount"],{n,10},{m,n}]//Column

Grid n×n

Entity["Graph",{"Grid",{n,n}}]["MinimalEdgeCoverCount"]

A288027

seq=Table[GraphData[{"Grid",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

Grid n×n×n

Entity["Graph",{"Grid",{n,n,n}}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Grid",{n,n,n}},"MinimalEdgeCoverCount"],{n,10}]

HalvedCube

seq=Table[GraphData[{"HalvedCube",n},"MinimalEdgeCoverCount"],{n,10}]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"MinimalEdgeCoverCount"],{n,5}]

Helm

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

Hypercube

Entity["Graph",{"Hypercube",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Hypercube",n},"MinimalEdgeCoverCount"],{n,0,10}]

Table[GraphData[{"Hypercube",n},"EdgeCoverCount"],{n,0,10}]

Keller

Entity["Graph",{"Keller",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Keller",n},"MinimalEdgeCoverCount"],{n,5}]

King m×n

King n×n

seq=Table[GraphData[{"King",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

Knight m×n

Block[{m,n},Sort[{{m,n}="Knight"/.GraphData[#,"NotationRules"],GraphData[#,"MinimalEdgeCoverCount"],XXX}&/@GraphData["Knight"]]]

DeleteCases[%,{_,x_,x_}]

Knight n×n

Entity["Graph",{"Knight",{n,n}}]["MinimalEdgeCoverCount"]

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

Ladder

Entity["Graph",{"Ladder",n}]["MinimalEdgeCoverCount"]

A288029

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

seq={1,2,6,17,45,120,324,873,2349,6322,17018,45809,123305,331904,893400,2404801,6473097,17423890,46900574,126244129,339816309,914696984,2462126012,6627401865,17839239445,48018585634,129253524146,347916817697,936501444241,2520817938240};

coef=FindLinearRecurrence[DeleteMissing@seq]

LinearRecurrence[{2,1,2,1,0,-1},{1,2,6,17,45,120},20]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

ToString[gf,InputForm]

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

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

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

Table[(2+I) (((-I))^n)+(2-I) (I^n),{n,10}]

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

Table[(1/10) ((2+I) (((-I))^n)+(2-I) (I^n)+2 (((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,1])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,1]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,2])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,2]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,3])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,3]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,4])^n) Root[125-850 #1+2120 ((#1)^2)-2184 ((#1)^3)+728 ((#1)^4)&,4])),{n,20}]//Round

RootSum[1-2 #^2-2 #^3+#^4&,#^n(47/91+33/182#-1/26#^2+4/91#^3)&]

2/182

FullSimplify[((2+I)(-I)^n+(2-I)I^n+ RootSum[1-2 ((#)^2)-2 ((#)^3)+((#)^4)&,94 ((#)^n)+33 ((#)^(n+1))-7 ((#)^(n+2))+8 ((#)^(n+3))&]/91)/10]

Table[(RootSum[1-2 ((#)^2)-2 ((#)^3)+((#)^4)&,94 ((#)^n)+33 ((#)^(n+1))-7 ((#)^(n+2))+8 ((#)^(n+3))&]+182 (2 Cos[((n π)/2)]+Sin[((n π)/2)]))/910,{n,20}]

LadderRung

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

MengerSponge

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

MoebiusLadder

Entity["Graph",{"MoebiusLadder",n}]["MinimalEdgeCoverCount"]

A290470

seq=Table[GraphData[{"MoebiusLadder",n},"MinimalEdgeCoverCount"],{n,3,25}]

seq={3,7,15,59,143,367,1039,2755,7395,20007,53727,144635,389535,1048159,2821535,7595267,20443523,55029319,148125295,398712379,1073232175,2888862159,7776059055,20931132355,56341155043,151655701607,408217663167,1098815603707,2957725352255};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{1,2,6,2,2,-2,-2,-1,1},{3,7,15,59,143,367,1039,2755,7395},20]

FindSequenceFunction[DeleteMissing@seq,n]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

((x (-3-4 x-2 (x^2)-12 (x^3)-6 (x^4)+4 (x^5)+6 (x^6)+4 (x^7)-3 (x^8)))/(-1+x+2 (x^2)+6 (x^3)+2 (x^4)+2 (x^5)-2 (x^6)-2 (x^7)-(x^8)+(x^9)))//Factor

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

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

RootReduce[res]

FullSimplify[(((-I))^n)+(I^n)]

Factor[(1/(2 (Sqrt[11])))I(2 I (Sqrt[11]) ((Root[-1+#1+((#1)^2)+((#1)^3)&,1])^n)+2 I (Sqrt[11]) ((Root[-1+#1+((#1)^2)+((#1)^3)&,2])^n)+2 I (Sqrt[11]) ((Root[-1+#1+((#1)^2)+((#1)^3)&,3])^n))]

terms=List@@ (((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,1])^(1+n)) Root[1+4 #1-8 ((#1)^2)+16 ((#1)^4)&,1]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,2])^(1+n)) Root[1+4 #1-8 ((#1)^2)+16 ((#1)^4)&,2]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,3])^(1+n)) Root[1+4 #1-8 ((#1)^2)+16 ((#1)^4)&,3]+((Root[1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,4])^(1+n)) Root[1+4 #1-8 ((#1)^2)+16 ((#1)^4)&,4]);

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

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

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

Mycielski

Entity["Graph",{"Mycielski",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Mycielski",n},"MinimalEdgeCoverCount"],{n,11}]

FindSequenceFunction[DeleteMissing@seq,n]

Odd

Entity["Graph",{"Odd",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Odd",n},"MinimalEdgeCoverCount"],{n,7}]

FindSequenceFunction[DeleteMissing@seq,n]

Pan

Padovan numbers

A000931

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[{1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65},n]

Table[RootSum[-1-#+((#)^3)&,-11 ((#)^(n+7))+4 ((#)^(n+8))+5 ((#)^(n+9))&]/23,{n,3,20}]

Path

Entity["Graph",{"Path",n}]["MinimalEdgeCoverCount"]

Padovan numbers

A000931

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

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-1]

Table[RootSum[-1-#+((#)^3)&,-11 ((#)^(n+4))+4 ((#)^(n+5))+5 ((#)^(n+6))&]/23,{n,20}]

PathComplement

A351588

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

Do[Print[{k,FindLinearRecurrence[Drop[DeleteMissing@seq,k]]//Head//Timing}],{k,0,5}]

Do[Print[{k,FindSequenceFunction[Drop[DeleteMissing@seq,k],n]//Head//Timing}],{k,0,5}]

Do[Print[{k,FindGeneratingFunction[Drop[DeleteMissing@seq,k],x]//Head//Timing}],{k,0,5}]

Prism

Entity["Graph",{"Prism",n}]["MinimalEdgeCoverCount"]

A290700

seq=Table[GraphData[{"Prism",n},"MinimalEdgeCoverCount"],{n,3,25}]

Table[GraphData[{"Prism",n},"EdgeCoverCount"],{n,3,25}]

seq={1,5,25,49,141,389,1009,2761,7441,19925,53769,144721,389325,1048325,2821665,7594761,20444065,55029413,148124153,398713969,1073231821,2888859781,7776063377,20931130057,56341150641,151655712629,408217654249,1098815597201};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length@coef]

LinearRecurrence[{1,2,6,2,2,-2,-2,-1,1},{1,5,25,49,141,389,1009,2761,7441},20]

FindSequenceFunction[DeleteMissing@seq,n]

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

CoefficientList[Series[-(( (1+4 x+18 (x^2)+8 (x^3)+10 (x^4)-12 (x^5)-14 (x^6)-8 (x^7)+9 (x^8)))/((1+(x^2)) (-1-x-(x^2)+(x^3)) (1-2 x-2 (x^2)+(x^4)))),{x,0,20}],x]

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

FullSimplify[RootReduce[res],n∈Integers&&n>0]

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

pos=(terms/.n->-1)[[All,2]]

f1=1-2 ((#1)^2)-2 ((#1)^3)+((#1)^4)&;
f2=1-2 #1-2 ((#1)^2)+((#1)^4)&;

Table[2 Cos[((n π)/2)]+RootSum[-1+#1+((#1)^2)+((#1)^3)&,#^n&]+Sum[((Root[f1,k])^(n+1))Root[f2,pos[[k]]],{k,Length[pos]}],{n,10}]//N//Chop

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

Equal@@Last/@an

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

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

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"MinimalEdgeCoverCount"],{n,10}]

SierpinskiCarpet

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

SierpinskiSieve

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimalEdgeCoverCount"],{n,5}]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"MinimalEdgeCoverCount"],{n,6}]

Star

Entity["Graph",{"Star",n}]["MinimalEdgeCoverCount"]

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

GraphData[{"Star",4},"MinimalEdgeCovers"]

GraphData[{"Empty",1},"MinimalEdgeCovers"]

Sun

A290763

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

seq={17,56,207,839,3579,16124,76037,373772,1907842,10080307,54988156,308997810,1785241070,10586718392,64343528516,400271482199,2545649131486,16533901290930,109563921896553,740108482190948,5092272608657314,35661352536071043};

Length[seq]

FindLinearRecurrence[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n-2]

gf=x FindGeneratingFunction[seq,x]

Clear[a,b]

Sum[n Binomial[j+k-1,j]Binomial[n-k-1,2k+2j-1]*(n-2k-j)^(j-i)/k,{k,1,(n-2j)/3}]

b[n_]:=b[n]=n!SeriesCoefficient[Exp[-x-x^2/2+x*Exp[x]],{x,0,n}]

a[n_]:=Sum[b[i]Sum[Binomial[j,i](2Binomial[n,2j](n-j)^(j-i)+Sum[n Binomial[j+k-1,j]Binomial[n-k-1,2k+2j-1]*(n-2k-j)^(j-i)/k,{k,1,(n-2j)/3}]),{j,i,n/2}],{i,0,n/2}]

Array[a,20,3]

Sunlet

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

Table[With[{gn={"Sunlet",n}},HighlightGraph[GraphData[gn,"Graph","Labeled"],GraphData[gn,"Edges"][[GraphData[gn,"MinimalEdgeCovers"][[1]]]]]],{n,5,7}]

GraphData[{"Sunlet",5},"MinimalEdgeCovers"]

GraphData[{"Sunlet",6},"MinimalEdgeCovers"]

GraphData[{"Sunlet",7},"MinimalEdgeCovers"]

Tetrahedral

seq=Table[GraphData[{"Tetrahedral",n},"MinimalEdgeCoverCount"],{n,6,10}]

Transposition

Entity["Graph",{"Transposition",n}]["MinimalEdgeCoverCount"]

seq=Table[GraphData[{"Transposition",n},"MinimalEdgeCoverCount"],{n,6}]

Triangular

seq=Table[GraphData[{"Triangular",n},"MinimalEdgeCoverCount"],{n,2,10}]

FindSequenceFunction[DeleteMissing@seq,n-1]

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"MinimalEdgeCoverCount"],{n,0,10}]

FindSequenceFunction[DeleteMissing@seq,n-1]

TriangularHoneycombAcuteKnight

A351591

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimalEdgeCoverCount"],{n,10}]

Do[Print[{k,FindLinearRecurrence[Drop[DeleteMissing@seq,k]]//Head//Timing}],{k,0,5}]

Do[Print[{k,FindSequenceFunction[Drop[DeleteMissing@seq,k],n]//Head//Timing}],{k,0,5}]

Do[Print[{k,FindGeneratingFunction[Drop[DeleteMissing@seq,k],x]//Head//Timing}],{k,0,5}]

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimalEdgeCoverCount"],{n,9}]

FindSequenceFunction[DeleteMissing@seq,n]

TriangularHoneycombObtuseKnight

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"MinimalEdgeCoverCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"MinimalEdgeCoverCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimalEdgeCoverCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

TriangularSnake

A308589

seq=Table[GraphData[{"TriangularSnake",n},"MinimalEdgeCoverCount"],{n,1,19,2}]

FindSequenceFunction[DeleteMissing@seq,n]

InputForm[%]

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

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

coeffs=an[[All,-1]]

SameQ@@coeffs

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

Table[-RootSum[-1-#1-2 ((#1)^2)+((#1)^3)&,-43 ((#1)^n)-99 ((#1)^(n+1))+40 ((#1)^(n+2))&]/87,{n,20}]

coef=FindLinearRecurrence[DeleteMissing@seq]

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

FindSequenceFunction[seq,n]

gf=x Factor@FindGeneratingFunction[seq,x]

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

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

InputForm[%]

Web

A001644

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

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

Wheel

Entity["Graph",{"Wheel",n}]["MinimalEdgeCoverCount"]

Table[-RootSum[-1-#1+((#1)^3)&,-((#1)^(1+n))+((#1)^(3+n))&]+(1/23) n RootSum[-1-#1+((#1)^3)&,-((#1)^(1+n))-8 ((#1)^(2+n))+13 ((#1)^(3+n))&]+RootSum[-1-#1-((#1)^2)+((#1)^3)&,((#1)^n)&],{n,4,20}]

A290471

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

seq={1,5,7,17,31,52,99,169,301,536,947,1698,3043,5482,9919,17993,32743,59713,109099,199634,365719,670605,1230547,2259286,4149841,7624904,14013499,25759764,47358741,87077529,160120891,294453673,541509775,995888884,1831584239};

coef=FindLinearRecurrence[DeleteMissing@seq]

Take[seq,Length[coef]]

LinearRecurrence[{1,5,7,17,31,52,99,169,301},{1,3,1,-5,-5,0,4,3,1},20]

FindSequenceFunction[DeleteMissing@seq,n-3]

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

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

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

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

RootReduce[res]//FullSimplify

f1=-1-#1+((#1)^3)&;
f2=148035889-529 ((#1)^2)+((#1)^3)&;
pos=Range[3];

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

Equal@@Last/@an

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

Table[-529 RootSum[-1-#1+((#1)^3)&,-((#1)^(n+1))+((#1)^(n+3))&],{n,10}]

Table[((Root[-1-#1+((#1)^3)&,1])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,1]+((Root[-1-#1+((#1)^3)&,2])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,3])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,3],{n,10}]//N//Chop

f1=-1-#1+((#1)^3)&;
f2=-32181715+194672 #1-529 ((#1)^2)+((#1)^3)&;
pos={1,3,2};

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

Equal@@Last/@an

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

Table[23 RootSum[-1-#1+((#1)^3)&,-((#1)^n)-8 ((#1)^(1+n))+13 ((#1)^(2+n))&],{n,10}]

Table[((Root[-1-#1+((#1)^3)&,3])^(1+n)) Root[-32181715+194672 #1-529 ((#1)^2)+((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,2])^(1+n)) Root[-32181715+194672 #1-529 ((#1)^2)+((#1)^3)&,3]+((Root[-1-#1+((#1)^3)&,1])^(2+n)) Root[-32181715+24334 #1+529 ((#1)^2)+((#1)^3)&,1],{n,10}]//N//Chop

Table[(1/529) (((Root[-1-#1+((#1)^3)&,1])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,1]+((Root[-1-#1+((#1)^3)&,2])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,3])^(1+n)) Root[148035889-529 ((#1)^2)+((#1)^3)&,3]+529 (((Root[-1-#1-((#1)^2)+((#1)^3)&,1])^n)+((Root[-1-#1-((#1)^2)+((#1)^3)&,2])^n)+((Root[-1-#1-((#1)^2)+((#1)^3)&,3])^n))+n (((Root[-1-#1+((#1)^3)&,3])^(1+n)) Root[-32181715+194672 #1-529 ((#1)^2)+((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,2])^(1+n)) Root[-32181715+194672 #1-529 ((#1)^2)+((#1)^3)&,3]+((Root[-1-#1+((#1)^3)&,1])^(2+n)) Root[-32181715+24334 #1+529 ((#1)^2)+((#1)^3)&,1])),{n,10}]//Round

Table[(1/529) (-529 RootSum[-1-#1+((#1)^3)&,-((#1)^(n+1))+((#1)^(n+3))&]+529 (RootSum[-1-#1-((#1)^2)+((#1)^3)&,((#)^n)&])+n 23 RootSum[-1-#1+((#1)^3)&,-((#1)^(n+1))-8 ((#1)^(n+2))+13 ((#1)^(n+3))&]),{n,10}]

FullSimplify[(1/529) (-529 RootSum[-1-#1+((#1)^3)&,-((#1)^(n+1))+((#1)^(n+3))&]+529 (RootSum[-1-#1-((#1)^2)+((#1)^3)&,((#)^n)&])+n 23 RootSum[-1-#1+((#1)^3)&,-((#1)^(n+1))-8 ((#1)^(n+2))+13 ((#1)^(n+3))&])]

Table[RootSum[-1-#-((#)^2)+((#)^3)&,((#)^n)&]-RootSum[-1-#+((#)^3)&,-((#)^(1+n))+((#)^(3+n))&]+ n RootSum[-1-#+((#)^3)&,-((#)^(n+1))-8 ((#)^(n+2))+13 ((#)^(n+3))&]/23,{n,10}]

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimalEdgeCoverCount"],{n,2,10}]