Two-Regular Graph

Author

Eric W. Weisstein
May 16, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Two-RegularGraph.nb.

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

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

Counts

Connected

Cycles only

Table[Total[Boole[TwoRegularGraphQ/@ConnectedGraphs[n]]],{n,10}]//Timing

Table[Length@GraphData[{"TwoRegular","Connected"},n],{n,12}]

Length/@Table[RegularConnectedGraphs[2,n],{n,12}]//Timing

All

A008483

Table[Total[Boole[TwoRegularGraphQ/@Graphs[n]]],{n,10}]//Timing

Length/@Table[GraphData["TwoRegular",n],{n,20}]

Table

GraphData["TwoRegular"]

Length/@Table[GraphData["TwoRegular",n],{n,10}]

Grid[Table[{n,GraphData["TwoRegular",n]},{n,10}],Alignment->Left,Dividers->All]

Formula

{1,-1,-1,1}.(#&/@(PartitionsP[n-#]&/@Range[0,3]))

Table[PartitionsP[-3+n]-PartitionsP[-2+n]-PartitionsP[-1+n]+PartitionsP[n],{n,0,12}]

FullSimplify[PartitionsP[-3+n]-PartitionsP[-2+n]-PartitionsP[-1+n]+PartitionsP[n]]//TraditionalForm

Product[(1/(1-(x^m))),{m,3,∞}]-1//FullSimplify

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

CoefficientList[Series[(((((-1+x))^2) (1+x))/(QPochhammer[x,x])),{x,0,12}],x]

With[{max=20},CoefficientList[Series[(((((-1+x))^2) (1+x))/(QPochhammer[x,x])),{x,0,max}],x]-Table[PartitionsP[-3+n]-PartitionsP[-2+n]-PartitionsP[-1+n]+PartitionsP[n],{n,0,max}]]//Union

Figure

GraphicsGrid[Map[If[#==="","",GraphData[#,"Image"]]&,Transpose[PadRight[#,5,""]&/@Table[GraphData["TwoRegular",n],{n,3,10}]],{2}]/.AbsolutePointSize[_]:>AbsolutePointSize[4],ImageSize->400,Dividers->All]

Checks

DeleteCases[{#,GraphData[#,"TwoRegular"],ContainsExactly[VertexDegree[GraphData[#]],{2}]}&/@GraphData[All],{_,x_,x_}]//Timing