Suzuki Graph

Author

Eric W. Weisstein
November 9, 2010

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

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Description

Strongly regular graph on 1782 vertices with parameters (v,k,lb,mu) = (1782,416,100,96) and group Suz.2.

Image

Graph["SuzukiGraph"]

GraphData["SuzukiGraph"]

Construction

Due to A.E. Brouwer
June 8, 2009

gap> LoadPackage("Grape");
gap> G:=PrimitiveGroup(1782,1);
gap> Size(G);
448345497600
gap> gamma:=NullGraph(G);;
gap> AddEdgeOrbit(gamma,[1,2]);
gap> IsDistanceRegular(gamma);
true
gap> CollapsedAdjacencyMat(gamma);
[ [ 0, 416, 0 ], [ 1, 100, 315 ], [ 0, 96, 320 ] ]
gap> GlobalParameters(gamma);
[ [ 0, 0, 416 ], [ 1, 100, 315 ], [ 96, 320, 0 ] ]
gap> G2:=AutomorphismGroup(gamma);
<permutation group with 8 generators>
gap> Size(G2);
896690995200

gamma is the Suzuki graph

{416,315;1,96}?

Check Automorphisms

Length[e]

g=FromUnorderedPairs[e]

FastAutomorphismCount[g]

FullForm[%]

Properties

g=Graph["SuzukiGraph"]

StronglyRegularQ[g]//Timing

RegularParameters[g]//Timing

DistanceRegularQ[g]//Timing

...ran overnight...

IntersectionArray[g]//Timing

... ran overnight ...

IntegerSpectrum[g]//SpectrumForm//TraditionalForm

FindHamiltonianCycle[g]//Timing

GraphData

GraphClassString

GraphClassString[{"SuzukiGraph"}]

GraphDataString

file=OpenWrite["suz.m"]

Factor[Times@@((x-#)&/@Join[Table[-16,{1001}],Table[20,{780}],Table[416,{1}]])]

WriteString[file,GraphDataString[g,
"Name"->"Suzuki graph",
"StandardName"->"SuzukiGraph",
"Vertices"->None,
"Classes"->{"Integral","Regular"},
"Spectrum"->Join[Table[-16,{1001}],Table[20,{780}],Table[416,{1}]],
"Skip"->{"Hamiltonian","HamiltonianCycles","HamiltonianPaths","StronglyRegular"},
"NotationRules"->{"StronglyRegular"->Flatten[{1782,{416},{100},{96}}]},
Debug->True,
TimeConstraint->10
]]//Timing

Close[file]