Octic Graph

Author

Eric W. Weisstein
May 18, 2017

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

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

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

Count

Connected

A014378

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

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

Length/@Table[RegularConnectedGraphs[8,n],{n,13}]

Hamiltonian

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

Length/@Table[Select[RegularConnectedGraphs[8,n],HamiltonianGraphQ],{n,13}]

Same as connected up up at least 13 nodes

Disconnected

A165878

A165878={0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,7,100,10901,3470736,1473822243,734843169811};

Position[A165878,Except[0,_Integer]]//Flatten

Table[Length@GraphData[{"Disconnected","Octic"},n],{n,21}]

Take[A165878,21]-Table[Length@GraphData[{"Disconnected","Octic"},n],{n,21}]

All

A180260

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

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

Figure

octic=GraphData["Octic"]

nice=SortBy[DeleteCases[octic,{"Octic"|"Circulant",_}],GraphData[#,"V"]&]

TableForm[GraphData[#,{"Name","V"}]&/@nice,TableDepth->2]

nicer=Select[nice,GraphData[#,"Embeddings"]=!={}&]

GraphicsGrid[Partition[GraphPlot[Graph[#],Method->None,BaseStyle->{FontSlant->"Italic",FontSize->9},PlotLabel->TextCell[GraphData[#,"Name"],PageWidth->100,TextAlignment->Center]]&/@nicer,4,4,{1,1},{}],ImageSize->400,Spacings->{-45,20}]

GraphData

GraphDataString

file=OpenWrite["~/octic.m"]

new=Cases[ToEntity/@Select[RegularConnectedGraphs[8,11],HamiltonianGraphQ],ToEntity[g_]:>g]//Quiet

Do[WriteString[file,GraphDataString[new[[i]],
"Embeddings"->{},
"EmbeddingClasses"->{},
"StandardName"->{"Octic",{11,i}},
"Name"->"11\[Hyphen]octic graph "<>ToString[i],
"Notation"->Missing["NotAvailable"],
"Information"->"OcticGraph",
Debug->True,
TimeConstraint->10
]],{i,Length[new]}];

Close[file]