Zero-Symmetric Graph

Author

Eric W. Weisstein
November 23, 2021

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Zero-SymmetricGraph.nb.

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

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

Counts

All

Total[Boole[Get[#]]]&/@FileNames["*.m",FileNameJoin[{$GraphDataDirectory,"Nodes","ZeroSymmetric"}]]

So smallest is on > 10 nodes (in fact, it is on 20 nodes)

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

Connected

Do[Print[{n,Total[Boole[ZeroSymmetricGraphQ/@ConnectedGraphs[n]]]//Timing}],{n,10}]

Table[Length@GraphData[{"ZeroSymmetric","Connected"},n],{n,20}]

Smallest Zero-Symmetric Graph

ZeroSymmetricGraph:=MakeUndirected[MakeGraph[Range[18],Mod[#1-#2,18]==1||(-1)^#1Mod[#1-#2,18]==5&]]

Figure

ShowGraph[ZeroSymmetricGraph]

Enumeration

PartitionLength[g_System`Graph]:=CountDistinct[CanonicalGraph[EdgeDelete[g,#]]&/@EdgeList[g]]

Length[l=GraphData[{"Connected","Cubic","VertexTransitive"}]]

Partition length 1

SortBy[Select[l,PartitionLength[#]==1&],GraphData[#,"V"]&]//Timing

Partition length 2

SortBy[Select[l,PartitionLength[#]==2&],GraphData[#,"V"]&]//Timing

Partition length 3

(zs=Monitor[Select[l,PartitionLength[GraphData[g=#]]===3&],g])//Timing

GraphicsGrid[Partition[GraphPlot[Graph[#],Method->None,PlotLabel->TextCell[GraphData[#,"Name"],PageWidth->100,TextAlignment->Center]]&/@zs,5,5,{1,1},{}],Spacings->{-10,40},ImageSize->600,BaseStyle->{FontSlant->"Italic"}]