EdgeCount

Author

Eric W. Weisstein
June 30, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Counts

(tallies=Tally[Sort[#]][[All,2]]&/@(values=Table[EdgeCount/@Graphs[n],{n,7}]))//Column

SetDirectory["/Users/eww/Documents/Mathematica/Paclets/GraphData/Nodes/EdgeCount"]

(tallies=Tally[Sort[#]][[All,2]]&/@(values=Get/@FileNames[]))//Column

A008406

Flatten[tallies]

Table[CoefficientList[GraphPolynomial[n,x],x],{n,8}]//Column

Table[NumberOfGraphs[v,e],{v,8},{e,0,Binomial[v,2]}]//Column

A000088

Total/@tallies

A086314

Total/@values

Table[D[GraphPolynomial[n,x],x],{n,18}]/.x->1

Table[Binomial[n,2]NumberOfGraphs[n]/2,{n,18}]

StringLength[ToString[%]]-2

Parametrized families

GraphDataCountTable["EdgeCount",{"LucasCube"->099920,"Pell"->364553}]

Classes

Pell

A364553

seq=Table[GraphData[{"Pell",n},"EdgeCount"],{n,0,9}]

FindSequenceFunction[seq,n+1]//FullSimplify

FindLinearRecurrence[seq]

LinearRecurrence[{4,-2,-4,-1},{0,1,5,18},30]

StringLength[ToString[%]]

gf=FindGeneratingFunction[seq,x]

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

EdgeCount/@pellgraphs

FindLinearRecurrence[%]

FindGeneratingFunction[%122,x]

seq={0,1,5,18,58,175,507,1428,3940,10701,28705,76230,200766};

gf=FindGeneratingFunction[seq,x]

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

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

(1/8)Table[ (-(((1-(Sqrt[2])))^n) (-2+(Sqrt[2]))+(((1+(Sqrt[2])))^n) (2+(Sqrt[2]))) n,{n,0,10}]//Expand

(1/8)Table[ ((-(((1-(Sqrt[2])))^n) (-2)-(((1-(Sqrt[2])))^n) (Sqrt[2]))+(((1+(Sqrt[2])))^n) (2)+(Sqrt[2])(((1+(Sqrt[2])))^n) ) n,{n,0,10}]//Expand

(1/8)Table[((Sqrt[2])((((1+(Sqrt[2])))^n)-(((1-(Sqrt[2])))^n))+2((((1-(Sqrt[2])))^n)+(((1+(Sqrt[2])))^n) ))n,{n,0,10}]//Expand

A000129

0,1,2,5,12

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

A001333

1, 1, 3, 7, 17, 41, 99,

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

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

(1/8)Table[(4Fibonacci[n,2]+4(-I)^n ChebyshevT[n,I])n,{n,0,10}]//Expand

Table[n(Fibonacci[n,2]+(-I)^n ChebyshevT[n,I])/2,{n,0,20}]