Hall-Janko Near-Octagon

Author

Eric W. Weisstein
October 17, 2018

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Hall-JankoNearOctagon.nb.

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

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

Embeddings

GraphData["HallJankoNearOctagon"]

Construction

Gap

HallJankoNearOctagon (315 vertices; Brouwer etal., p. 224)

From Andries Brouwer,
Fri, 12 Jun 2009 20:22:38 +0200
Sat, 13 Jun 2009 08:29:09 +0200

% gap
gap> G:=PrimitiveGroup(315,2);
J_2
gap> Size(G);
604800
gap> NrMovedPoints(G);
315

checking that this really is the desired permutation representation

gap> LoadPackage("grape");;
gap> gamma:=NullGraph(G);;
gap> S:=Stabilizer(G,1);
<permutation group of size 1920 with 2 generators>
gap> OrbitLengths(S);
[ 80, 160, 10, 32, 32 ]
gap> Orbits(S)[3];
[ 5, 249, 133, 19, 298, 194, 85, 97, 115, 296 ]
gap> AddEdgeOrbit(gamma,[1,5]);
gap> GlobalParameters(gamma);
[ [ 0, 0, 10 ], [ 1, 1, 8 ], [ 1, 1, 8 ], [ 4, 4, 2 ], [ 5, 5, 0 ] ]
gap> IsDistanceRegular(gamma);
true

This constructed the HallJanko graph on 315 vertices, not to be confused with the HallJanko graph on 100 vertices.  It carries a near octagon, with 525 lines: 5 lines of size 3 on each point,

gap> Intersection(Adjacency(gamma,1),Adjacency(gamma,9));
[ 225 ]
gap> Lines:=Orbit(G,[1,9,225],OnSets);;
gap> Size(Lines);
525

ReadGAPEdgeList[f_String]:=ToExpression[StringReplace[Import[f,"String"],{"["->"{","]"->"}"}]]

e=ReadGAPEdgeList["hj.gap"]

g=FromUnorderedPairs[e]

distanceregular.org

g=DistanceRegularGraph["halljanko315"]

ToEntity[g]

GraphData["Hall*"]

VertexCount[g]

GraphDiameter[g]//Timing

RegularParameters[g]//Timing

DistanceRegularGraphQ[g]//Timing

IntersectionArray[g]//Timing

IntegerSpectrum[g]//SpectrumForm

GroupOrder[GraphAutomorphismGroup[g]]

HamiltonianGraphQ[g]

Properties

StronglyRegularQ[g]//Timing

WeaklyRegularQ[g]//Timing

RegularParameters[g]//Timing

DistanceRegularQ[g]//Timing

IntersectionArray[g]//Timing

FindHamiltonianCycle[g]//Timing

TimeConstrained[HamiltonianCycles[g,1],60]//Timing

FastAutomorphismCount[g]//Timing

VertexTransitiveQ[Graph["HallJankoNearOctagon"]]//Timing

GetGraphData["HallJankoNearOctagon","File"]

FastAutomorphismCount[LineGraph[g]]//Timing

IntegerSpectrum[g]//SpectrumForm//TraditionalForm

LocalGraphQ[g]//Timing

LocalGraph[g]//RecognizeGraph

GraphData

GraphDataString

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

WriteString[file,GraphDataString[g,
"StandardName"->"HallJankoNearOctagon",
"Name"->"Hall-Janko near octagon",
"Vertices"->None,
"Information"->"Distance-TransitiveGraph",
"Classes"->{"DistanceRegular","DistanceTransitive","Integral","Regular","VertexTransitive","WeaklyRegular"},
"Skip"->{"ClawFree","EdgeTransitive","VertexTransitive"},
"NotationRules"->{"DistanceRegular"->{{10,8,8,2},{1,1,4,5}}},
Debug->True,
TimeConstraint->60
]]//Timing

Close[file]