Conway-Smith Graph

Author

Eric W. Weisstein
October 17, 2018

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Conway-SmithGraph.nb.

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

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

Embeddings

GraphData["ConwaySmithGraph","Graph","All"]

Construction

GAP

http://www.gap-system.org/oldsite/Forum/Soicher.1/Leonard.1/Re__Grap.4/1.html

g:=rec(
  isGraph := true,
  order := 63,
  group := Group( ( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18) (19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39) (40,41,42)(43,44,45)(46,47,48)(49,50,51)(52,53,54)(55,56,57)(58,59,60) (61,62,63), ( 1,31)( 2,33)( 3,32)( 4,44)( 5,43)( 6,45)( 7,54)( 8,53) ( 9,52)(10,58)(11,60)(12,59)(13,61)(14,63)(15,62)(17,18)(19,20)(22,24) (26,27)(29,30)(35,36)(37,39)(40,41)(46,47)(49,51)(56,57), ( 1,16)( 2,18) ( 3,17)( 4, 6)( 7, 8)(11,12)(14,15)(19,45)(20,44)(21,43)(22,53)(23,52) (24,54)(25,58)(26,60)(27,59)(28,61)(29,63)(30,62)(32,33)(35,36)(37,38) (40,42)(46,48)(49,50)(56,57), ( 2, 3)( 4,17)( 5,16)( 6,18)( 7, 8)(10,12) (13,14)(19,33)(20,32)(21,31)(22,24)(25,26)(28,30)(34,52)(35,54)(36,53) (37,58)(38,60)(39,59)(40,61)(41,63)(42,62)(44,45)(47,48)(50,51)(56,57), ( 2, 3)( 4, 6)( 7,18)( 8,17)( 9,16)(10,11)(13,15)(19,20)(22,32)(23,31) (24,33)(25,27)(28,29)(34,43)(35,45)(36,44)(38,39)(41,42)(46,58)(47,60) (48,59)(49,61)(50,63)(51,62)(53,54)(56,57), ( 2, 3)( 4, 5)( 7, 9)(10,16) (11,18)(12,17)(14,15)(19,21)(22,23)(25,31)(26,33)(27,32)(29,30)(35,36) (37,43)(38,45)(39,44)(41,42)(46,52)(47,54)(48,53)(50,51)(55,61)(56,63) (57,62)(59,60), ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17)(20,21) (23,24)(26,27)(28,31)(29,33)(30,32)(35,36)(38,39)(40,43)(41,45)(42,44) (47,48)(49,52)(50,54)(51,53)(55,58)(56,60)(57,59)(62,63) ),   schreierVector := [ -1, 1, 1, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 3, 1, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 2, 1, 2, 5, 5, 5, 6, 6, 6, 7, 7, 7, 3, 3, 3, 6, 6, 6, 7, 7, 7, 3, 3, 3, 7, 7, 7, 3, 3, 3, 3, 3, 3 ],   adjacencies := [ [ 34, 37, 40, 43, 46, 49, 52, 55, 58, 61 ] ],
  representatives := [ 1 ],
  isSimple := true,
  names := [ [ [ 1, 2 ], 1 ], [ [ 1, 2 ], 2 ], [ [ 1, 2 ], 3 ],  [ [ 1, 3 ], 1 ], [ [ 1, 3 ], 2 ], [ [ 1, 3 ], 3 ], [ [ 1, 4 ], 1 ], [ [ 1, 4 ], 2 ], [ [ 1, 4 ], 3 ], [ [ 1, 5 ], 1 ], [ [ 1, 5 ], 2 ], [ [ 1, 5 ], 3 ], [ [ 1, 6 ], 1 ], [ [ 1, 6 ], 2 ], [ [ 1, 6 ], 3 ], [ [ 1, 7 ], 1 ], [ [ 1, 7 ], 2 ], [ [ 1, 7 ], 3 ], [ [ 2, 3 ], 1 ], [ [ 2, 3 ], 2 ], [ [ 2, 3 ], 3 ], [ [ 2, 4 ], 1 ], [ [ 2, 4 ], 2 ], [ [ 2, 4 ], 3 ], [ [ 2, 5 ], 1 ], [ [ 2, 5 ], 2 ], [ [ 2, 5 ], 3 ], [ [ 2, 6 ], 1 ], [ [ 2, 6 ], 2 ], [ [ 2, 6 ], 3 ], [ [ 2, 7 ], 1 ], [ [ 2, 7 ], 2 ], [ [ 2, 7 ], 3 ], [ [ 3, 4 ], 1 ], [ [ 3, 4 ], 2 ], [ [ 3, 4 ], 3 ], [ [ 3, 5 ], 1 ], [ [ 3, 5 ], 2 ], [ [ 3, 5 ], 3 ], [ [ 3, 6 ], 1 ], [ [ 3, 6 ], 2 ], [ [ 3, 6 ], 3 ], [ [ 3, 7 ], 1 ], [ [ 3, 7 ], 2 ], [ [ 3, 7 ], 3 ], [ [ 4, 5 ], 1 ], [ [ 4, 5 ], 2 ], [ [ 4, 5 ], 3 ], [ [ 4, 6 ], 1 ], [ [ 4, 6 ], 2 ], [ [ 4, 6 ], 3 ], [ [ 4, 7 ], 1 ], [ [ 4, 7 ], 2 ], [ [ 4, 7 ], 3 ], [ [ 5, 6 ], 1 ], [ [ 5, 6 ], 2 ], [ [ 5, 6 ], 3 ], [ [ 5, 7 ], 1 ], [ [ 5, 7 ], 2 ], [ [ 5, 7 ], 3 ], [ [ 6, 7 ], 1 ], [ [ 6, 7 ], 2 ], [ [ 6, 7 ], 3 ] ] 
  );
  gap> IsDistanceRegular(g);
true
gap> GlobalParameters(g);
[ [ 0, 0, 10 ], [ 1, 3, 6 ], [ 2, 4, 4 ], [ 6, 3, 1 ], [ 10, 0, 0 ] ]
PrintTo(”~/cs.gap”,UndirectedEdges(g));

e=ReadGAPEdgeList["cs.gap"]

g=FromUnorderedPairs[e]

distanceregular.org

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

Properties

DistanceRegularQ[g]//Timing

IntersectionArray[g]//Timing

LocalGraphQ[g]//Timing

RecognizeGraph[LocalGraph[g]]

GraphPlot[g]

GraphData

GraphDataString

GraphDataString[g,
"StandardName"->"ConwaySmithGraph",
"Name"->"Conway-Smith graph",
"Vertices"->None,
"Information"->"Conway-SmithGraph",
TimeConstraint->60,
Debug->True
]