Quintic Symmetric Graph

Author

Eric W. Weisstein
May 18, 2017

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

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

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

Additional Authors

Snub dodecahedral graph embedding by
Ed Pegg Jr.
October 18, 2002

Counts

Connected

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

Table[Length@GraphData[{"Quintic","Connected"},n],{n,13}]

Length/@Table[RegularConnectedGraphs[5,n],{n,13}]

Hamiltonian

Table[Length@GraphData[{"Quintic","Hamiltonian"},n],{n,13}]

Length/@Table[Select[RegularConnectedGraphs[5,n],HamiltonianGraphQ],{n,13}]

Disconnected

Table[Length@GraphData[{"Quintic","Disconnected"},n],{n,13}]

All

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

Table[Length@GraphData["Quintic",n],{n,13}]

Enumeration

V5

Length/@(symquint=Table[Select[Graphs[n],QuinticQ[#]&&SymmetricGraphQ[#]&],{n,9}])//Timing

ShowGraphArray[RecognizeGraph/@Flatten[symquint],AspectRatio->Automatic]

list={
CompleteGraph[6],
IcosahedralGraph,
ClebschGraph,
SnubCubicalGraph,
SnubDodecahedralGraph
};

ShowGraphArray[Partition[list,4,4,{1,1},{}],
AspectRatio->Automatic,VertexStyle->Disk[.05]
]

V7

l=SortBy[GraphData[{"Quintic","Symmetric","Connected"}],GraphData[#,"V"]&]

Length[l]

(#[[1,1]]->#[[All,2]])&/@SplitBy[GraphData[#,{"V","#"}]&/@l,First]//Column

GraphicsGrid[Partition[Show[GraphData[#],PlotLabel->TextCell[GraphData[#,"Name"],PageWidth->70,TextAlignment->Center],BaseStyle->{Italic,FontSize->9}]&/@Drop[l,-1],5,5,{1,1},{}]/.AbsolutePointSize[_]:>AbsolutePointSize[4],ImageSize->400]