Doubly Truncated Witt Graph

Author

Eric W. Weisstein
October 17, 2018

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

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

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

Embeddings

Default

GraphData["DoublyTruncatedWittGraph","Graph","All"]

Construction

Golay

The doubly truncated Witt graph on 330 points is constructed by removing all vectors of the large Witt design containing any two arbitrarily chosen symbols. This is a \href{distance-regular graph} with intersection array {7, 6, 4, 4 ; 1, 1, 1, 6}.

Andries E. Brouwer
Mon, 8 Jun 2009 13:18:58 +0200

The large Witt graph has as vertices the 759 words of weight 8 in extended binary Golay. Call the octads, and view them as sets of size 8.  Pick one coordinate position. 253 octads have a 1 there, 506 octads have a 0 there.  I conjecture that your IofinovaIvanovGraph506 is the same as what I describe in [BCN] 11.4.2 - it is the induced subgraph on 506 vertices from the 759.

Pick a second coordinate position. Of the 506, there are 176 with a 1 there, and 330 with a 0. The induced subgraph from 759, or from 506 on this 330 gives you the DoublyTruncatedWittGraph

Length[w8=Select[ExtendedBinaryGolayCode,Total[#]==8&]]

(z=Flatten[Position[w8[[All,1]],0]])//Length

(LargeWitt=MakeGraph[w8,#1.#2==0&,Type->Undirected])

g506=InduceSubgraph[LargeWitt,z]

RecognizeGraph[%]

(z2=Flatten[Position[w8[[z]][[All,2]],0]])//Length

g330=InduceSubgraph[g506,z2]

RecognizeGraph[%]

FastAutomorphismCount[g330]

IntegerSpectrum[g330]//SpectrumForm//TraditionalForm

distanceregular.org

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/dtwitt.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["dtwitt.m"]

s=Join[Table[-4,{21}],Table[-3,{99}],Table[1,{154}],Table[4,{55}],Table[7,{1}]]

Times@@((x-#)&/@s)

WriteString[file,GraphDataString[g330,
"Name"->"doubly truncated Witt graph",
"StandardName"->"DoublyTruncatedWittGraph",
"Vertices"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Integral","Regular","WeaklyRegular"},
"NotationRules"->{"DistanceRegular"->{{7,6,4,4},{1,1,1,6}},"WeaklyRegular"->{330,{7},{0},{0,1}}},
"Spectrum"->s,
Debug->True,
TimeConstraint->60
]]//Timing

Close[file]