Doro Graph

Author

Eric W. Weisstein
October 17, 2018

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

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

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

Embeddings

Default

GraphData["DoroGraph","Graph","All"]

LCF

GraphData["DoroGraph","LCF"]

GraphData["DoroGraph","LCFNotations"]

Construction

GAP

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

These two lines name the same graph.
The group is PGammaL(2,16) = PSigmaL(2,16) on 68 vertices, valency 12.

LoadPackage("Grape");
gap> G:=PrimitiveGroup(68,3);
PSigmaL(2, 16)
gap> NrMovedPoints(G);
68
gap> S:=Stabilizer(G,1);
<permutation group of size 240 with 3 generators>
gap> gamma:=NullGraph(G);;
gap> OrbitLengths(S);
[ 40, 15, 12 ]
gap> Orbits(S)[3];
[ 6, 29, 55, 66, 36, 30, 15, 64, 32, 63, 41, 10 ]
gap> AddEdgeOrbit(gamma,[1,6]);
gap> Diameter(gamma);
3
gap> IsDistanceRegular(gamma);
true
gap> GlobalParameters(gamma);
[ [ 0, 0, 12 ], [ 1, 1, 10 ], [ 3, 6, 3 ], [ 8, 4, 0 ] ]
PrintTo("~/doro.gap",UndirectedEdges( gamma ));

That was the Doro graph.
You see that the orbitlengths 12, 40, 15 of S are the number of vertices at distance 1, 2, 3 of the fixed vertex 1.
That is always the case (by definition) for distance-transitive graphs.

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

g=FromUnorderedPairs[e]

distanceregular.org

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/doro68.am"]

ToEntity[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

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

WriteString[file,
GraphDataString[g,
"StandardName"->"DoroGraph",
"Name"->"Doro graph",
"Information"->{"DoroGraph"},
"Skip"->{"ChromaticPolynomial","ClawFree","IndependencePolynomial","MatchingPolynomial"},
"Classes"->{"DistanceTransitive","DistanceRegular","Integral"},
"NotationRules"->{"DistanceRegular"->{{12,10,3},{1,3,8}}},
Vertices->None,
Debug->True,
TimeConstraint->60
]]//Timing