Livingstone Graph

Author

Eric W. Weisstein
October 18, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

GraphData["LivingstoneGraph","Graph","All"]

Construction

GAP

Date: Sun, 14 Jun 2009 15:43:10 +0200                                                                     
From: Andries E. Brouwer

The group is J1 on 266 vertices, the valency is 11.

gap> LoadPackage("Grape");;
gap> G:=PrimitiveGroup(266,1);
J_1
gap> NrMovedPoints(G);
266
gap> S:=Stabilizer(G,1);
<permutation group of size 660 with 3 generators>
gap> gamma:=NullGraph(G);;
gap> OrbitLengths(S);
[ 12, 132, 11, 110 ]
gap> Orbits(S)[3];
[ 6, 59, 47, 69, 70, 72, 46, 44, 58, 45, 71 ]
gap> AddEdgeOrbit(gamma,[1,6]);
gap> Diameter(gamma);
4
gap> IsDistanceRegular(gamma);
true
gap> GlobalParameters(gamma);
[ [ 0, 0, 11 ], [ 1, 0, 10 ], [ 1, 4, 6 ], [ 5, 5, 1 ], [ 11, 0, 0 ] ]
PrintTo(”~/liv.gap”,UndirectedEdges( gamma ));

e=ReadGAPEdgeList["~/liv.gap"]

g=FromUnorderedPairs[e]

distanceregular.org

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/livingstone.am"]

RecognizeGraph[g]

VertexCount[g]

GraphDiameter[g]//Timing

RegularParameters[g]//Timing

DistanceRegularGraphQ[g]//Timing

IntersectionArray[g]//Timing

IntegerSpectrum[g]//SpectrumForm

GroupOrder[GraphAutomorphismGroup[g]]

HamiltonianGraphQ[g]

GraphData

GraphDataString

WriteString[file,
GraphDataString[g,
"StandardName"->"LivingstoneGraph",
"Name"->"Livingstone graph",
"Information"->{"LivingstoneGraph"},
"Skip"->{"ChromaticPolynomial","ClawFree","EdgeTransitive","IndependencePolynomial","MatchingPolynomial"},
"Classes"->{"DistanceTransitive","DistanceRegular","VertexTransitive"},
"NotationRules"->{"DistanceRegular"->{{11,10,6,1,1,1,5,11}}},
Vertices->None,
Debug->True,
TimeConstraint->60
]]//Timing

Close[file]