McLaughlin Graph

Author

Eric W. Weisstein
March 28, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

GraphData["McLaughlinGraph","Graph","All"]

Construction

Royle

http://www.csse.uwa.edu.au/~gordon/constructions/mclaughlin/index.html

long list of pairs initialization

McLaughlinGraph =FromUnorderedPairs[(1+{{0,23},{0,25},{0,26},{0,27},{0,28},{0,29},{0,31},{0,32},{0,33},{0,35},{0,36},{0,37},{0,39},{0,41},{0,45},{0,46},{0,49},{0,52},{0,54},{0,59},{0,61},{0,63},{0,65},{0,68},{0,69},{0,70},{0,76},{0,78},{0,80},{0,83},{0,85},{0,86},{0,87},{0,89},{0,91},{0,93},{0,94},{0,96},{0,97},{0,102},{0,103},{0,104},{0,107},{0,109},{0,117},{0,118},{0,119},{0,120},{0,122},{0,125},{0,128},{0,129},{0,133},{0,134},{0,140},{0,141},{0,142},{0,144},{0,145},{0,148},{0,149},{0,150},{0,153},{0,154},{0,155},{0,158},{0,159},{0,160},{0,161},{0,162},{0,165},{0,166},{0,168},{0,169},{0,173},{0,176},{0,177},{0,179},{0,181},{0,187},{0,190},{0,193},{0,196},{0,197},{0,198},{0,201},{0,204},{0,207},{0,208},{0,213},{0,214},{0,216},{0,222},{0,224},{0,231},{0,235},{0,237},{0,244},{0,245},{0,246},{0,248},{0,251},{0,254},{0,260},{0,263},{0,264},{0,265},{0,266},{0,267},{0,270},{0,271},{0,272},{1,22},{1,24},{1,26},{1,27},{1,30},{1,31},{1,32},{1,33},{1,34},{1,35},{1,38},{1,40},{1,41},{1,42},{1,43},{1,44},{1,45},{1,47},{1,48},{1,49},{1,50},{1,54},{1,56},{1,57},{1,59},{1,60},{1,61},{1,68},{1,69},{1,72},{1,74},{1,78},{1,79},{1,81},{1,83},{1,85},{1,86},{1,93},{1,98},{1,100},{1,103},{1,105},{1,108},{1,109},{1,113},{1,114},{1,115},{1,117},{1,119},{1,122},{1,125},{1,130},{1,131},{1,134},{1,136},{1,139},{1,141},{1,142},{1,148},{1,149},{1,150},{1,151},{1,152},{1,154},{1,156},{1,158},{1,162},{1,164},{1,168},{1,170},{1,171},{1,172},{1,180},{1,181},{1,184},{1,188},{1,192},{1,193},{1,196},{1,197},{1,209},{1,212},{1,213},{1,214},{1,215},{1,216},{1,218},{1,219},{1,222},{1,225},{1,227},{1,231},{1,233},{1,237},{1,242},{1,244},{1,245},{1,251},{1,254},{1,255},{1,256},{1,257},{1,259},{1,260},{1,262},{1,263},{1,264},{1,266},{1,267},{1,270},{1,272},{1,274},{2,22},{2,23},{2,24},{2,26},{2,27},{2,29},{2,30},{2,31},{2,36},{2,38},{2,39},{2,42},{2,43},{2,44},{2,51},{2,53},{2,55},{2,56},{2,58},{2,60},{2,63},{2,67},{2,69},{2,71},{2,72},{2,76},{2,78},{2,79},{2,86},{2,87},{2,90},{2,91},{2,92},{2,93},{2,102},{2,103},{2,107},{2,111},{2,114},{2,116},{2,117},{2,118},{2,122},{2,124},{2,126},{2,129},{2,130},{2,134},{2,136},{2,140},{2,144},{2,147},{2,148},{2,149},{2,152},{2,154},{2,155},{2,158},{2,163},{2,166},{2,167},{2,168},{2,175},{2,176},{2,177},{2,182},{2,183},{2,187},{2,190},{2,191},{2,192},{2,193},{2,195},{2,196},{2,197},{2,198},{2,199},{2,201},{2,203},{2,213},{2,214},{2,216},{2,220},{2,222},{2,223},{2,225},{2,226},{2,227},{2,229},{2,230},{2,231},{2,233},{2,235},{2,236},{2,237},{2,238},{2,242},{2,245},{2,246},{2,251},{2,253},{2,254},{2,255},{2,256},{2,260},{2,262},{2,263},{2,264},{2,268},{2,272},{2,273},{2,274},{3,22},{3,23},{3,25},{3,27},{3,28},{3,29},{3,30},{3,34},{3,36},{3,38},{3,39},{3,40},{3,42},{3,45},{3,46},{3,47},{3,50},{3,52},{3,53},{3,55},{3,59},{3,61},{3,62},{3,63},{3,64},{3,65},{3,66},{3,67},{3,68},{3,70},{3,71},{3,74},{3,76},{3,82},{3,85},{3,86},{3,90},{3,91},{3,93},{3,94},{3,96},{3,98},{3,101},{3,107},{3,109},{3,113},{3,114},{3,117},{3,119},{3,120},{3,127},{3,134},{3,135},{3,137},{3,142},{3,144},{3,147},{3,148},{3,149},{3,150},{3,151},{3,153},{3,154},{3,161},{3,162},{3,163},{3,168},{3,173},{3,174},{3,176},{3,178},{3,181},{3,186},{3,188},{3,190},{3,194},{3,196},{3,197},{3,202},{3,203},{3,206},{3,209},{3,210},{3,212},{3,213},{3,216},{3,222},{3,225},{3,227},{3,233},{3,234},{3,235},{3,236},{3,238},{3,240},{3,241},{3,242},{3,245},{3,247},{3,249},{3,251},{3,253},{3,254},{3,256},{3,257},{3,260},{3,262},{3,263},{3,270},{3,272},{3,273},{3,274},{4,23},{4,25},{4,26},{4,27},{4,29},{4,30},{4,34},{4,35},{4,37},{4,38},{4,40},{4,46},{4,47},{4,50},{4,51},{4,52},{4,53},{4,54},{4,56},{4,57},{4,58},{4,59},{4,64},{4,65},{4,66},{4,68},{4,69},{4,72},{4,73},{4,74},{4,75},{4,77},{4,79},{4,80},{4,81},{4,82},{4,83},{4,86},{4,89},{4,91},{4,94},{4,101},{4,103},{4,105},{4,112},{4,117},{4,119},{4,122},{4,124},{4,129},{4,130},{4,131},{4,133},{4,134},{4,138},{4,140},{4,141},{4,142},{4,144},{4,146},{4,149},{4,150},{4,154},{4,164},{4,166},{4,168},{4,169},{4,172},{4,174},{4,176},{4,177},{4,181},{4,184},{4,185},{4,189},{4,190},{4,192},{4,193},{4,196},{4,198},{4,199},{4,204},{4,205},{4,216},{4,217},{4,220},{4,221},{4,222},{4,223},{4,227},{4,231},{4,233},{4,235},{4,236},{4,237},{4,238},{4,247},{4,248},{4,249},{4,252},{4,253},{4,254},{4,256},{4,258},{4,262},{4,263},{4,266},{4,268},{4,270},{4,272},{4,273},{4,274},{5,23},{5,24},{5,25},{5,27},{5,29},{5,30},{5,33},{5,34},{5,35},{5,38},{5,39},{5,40},{5,48},{5,51},{5,52},{5,54},{5,57},{5,58},{5,59},{5,62},{5,64},{5,67},{5,75},{5,77},{5,78},{5,79},{5,84},{5,87},{5,88},{5,89},{5,90},{5,92},{5,96},{5,100},{5,103},{5,104},{5,105},{5,107},{5,109},{5,112},{5,114},{5,116},{5,117},{5,118},{5,119},{5,120},{5,122},{5,123},{5,125},{5,129},{5,130},{5,134},{5,135},{5,148},{5,150},{5,151},{5,152},{5,154},{5,157},{5,158},{5,159},{5,162},{5,166},{5,168},{5,169},{5,173},{5,175},{5,176},{5,180},{5,181},{5,183},{5,185},{5,188},{5,190},{5,192},{5,193},{5,195},{5,197},{5,198},{5,200},{5,203},{5,208},{5,209},{5,210},{5,213},{5,214},{5,215},{5,222},{5,224},{5,227},{5,2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Tests

Map[{First[#], Length[#]} &, Split[Sort[DegreeSequence[McLaughlinGraph]]]]

Through[{V,M}[McLaughlinGraph]]

DegreeSequence[McLaughlinGraph]

Witt Design (NOT WORKING)

Consider as vertices the 253 length-7 vectors v and 23 points p. Set edges such that p&v are adjacent if $p \notin v$, and $v_j \& v_k$ are adjacent if they share a single term. Select an arbitrary vertex.  For all 176 neighbors of that vertex, change edges to non-edges, and non-edges to edges. Eliminate the now isolated vertex, and the remaining 275 vertex graph is the \href{McLaughlin graph}.

Gordon Royle
- take the 253 blocks of the Witt design (a 4-(23,7,1) design)
- form a 276 vertex graph using these 253 blocks together with the points 1..23
where (a) two blocks are adjacent if they meet in one point and (b) a point and
a block are adjacent if the block does NOT contain the point.

Now this graph has 276 vertices, but is not regular, so we continue..

- pick some fixed vertex v (any vertex will do) and identify its neighbours
N(v)
- switch on N(v); this means that any edge with one end in N(v) and one end out
of N(v) gets turned to a non-edge, while any non-edge with one end in N(v) gets
turned to an edge

This graph now consists of an isolated vertex (v) together with a regular
275-vertex graph called the McLaughlin graph, so the final step is just to
throw away the isolated vertex...

McLaughlin1 =Sort[Drop[
Flatten[Table[If[Length[Union[Flatten[{WittLength7[[mm]],WittLength7[[nn]]}]]]==13,{277-mm,277-nn},0],{mm,1,252},{nn,mm+1,253}],1]//Union,1]];
McLaughlin2=Sort[Drop[
Flatten[Table[If[Length[Union[Flatten[{WittLength7[[mm]],{nn}}]]]==8,{277-mm,nn},0],{mm,1,253},{nn,1,23}],1]//Union,1]];
McLaughlin3 = FromUnorderedPairs[Sort[Sort/@Join[McLaughlin1,McLaughlin2]]];
Map[{First[#],Length[#]}&,Split[Sort[DegreeSequence[McLaughlin3]]]]

Take[Sort[Sort/@Join[McLaughlin1,McLaughlin2]],176]

jj = ToAdjacencyMatrix[McLaughlin3];
kk = Flatten[Position[First[jj],1]];

SwitchM3 = Table[Mod[jj[[a,b]]+If[a≠ b&&(MemberQ[kk,a]||MemberQ[kk,b]),1,0],2],{a,1,Length[jj]},{b,1,Length[jj]}];

McLaughlinGraph = Drop[Drop[#,1]&/@With[{jj = ToAdjacencyMatrix[McLaughlin3]},
With[{kk = Flatten[Position[First[jj],1]]},
Table[Mod[jj[[a,b]]+If[a≠ b&&(MemberQ[kk,a]||MemberQ[kk,b]),1,0],2],{a,1,Length[jj]},{b,1,Length[jj]}]]],1]//FromAdjacencyMatrix;
Map[{First[#],Length[#]}&,Split[Sort[DegreeSequence[McLaughlinGraph]]]]

distanceregular.org

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

(2, 2, (Sqrt[6])) Leech lattice construction

https://cp4space.hatsya.com/2013/09/12/leech-lattice/

Start with three Leech lattice points that form the vertices of an isosceles triangle with side lengths of 2, 2, and sqrt(6), identifying the exactly 275 lattice points that are at distance 2 from each triangle vertex, and joining two points with an edge if they are separated by distance sqrt(6).

From Michael Trott, Marc 28, 2022

basis=LatticeData["Leech","Basis"]//Normal;

triangle=SelectFirst[Subsets[basis,{3}],Sort[EuclideanDistance@@@Subsets[#,{2}]]=={2,2,Sqrt[6]}&]

{EuclideanDistance@@triangle[[{1,2}]],
EuclideanDistance@@triangle[[{1,3}]],
EuclideanDistance@@triangle[[{2,3}]]}

triangleN=Developer`ToPackedArray/@N[triangle];
basisN=Developer`ToPackedArray/@N[basis] ;
distances[p_] :=  EuclideanDistance[#,p]&/@ triangle

bag

coefficients=;

Length[Union[coefficients]]

allPoints=Together[Total[basis #]]&/@ coefficients;

Union[distances/@ allPoints//FullSimplify]

Norm/@ allPoints//Tally

ArrayPlot[Transpose[Sort@allPoints]]

Length[ee=Select[Subsets[Range[275],{2}],Abs[SquaredEuclideanDistance@@allPoints[[#]]-6]<.01&]]//Quiet

g=Graph[UndirectedEdge@@@ee]

RecognizeGraph[g]

Subconstituents

First = GQ(3,9)

g1=DistanceRegularGraph["http://www.distanceregular.org/graphdata/1sub-mclaughlingraph.am"]

RecognizeGraph[g1]

Second

IsomorphicGraphQ[GraphData["LocalMcLaughlinGraph"],g2]

GraphData["LocalMcLaughlinGraph","NotationRules"]

g2=DistanceRegularGraph["http://www.distanceregular.org/graphdata/2sub-mclaughlingraph.am"]

RecognizeGraph[g2]

GraphData

GraphClassString

GraphClassString["McLaughlinGraph"]

GraphDataString

VertexCount[g2]

file=OpenWrite["~/mcl2.m"]

WriteString[file,GraphDataString[g2,
"StandardName"->"McLaughlinGraphSubconstituent2",
"Name"->"McLaughlin graph second subconstituent",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid"},
Debug->True,
TimeConstraint->10]]

Close[file]