Schläfli Graph

Author

Eric W. Weisstein
October 26, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

<<MathWorld`Graphs`

Names

GraphData["SchlaefliGraph","StandardName"]

GraphData["SchlaefliGraph","StandardNames"]

Primary

GraphData["SchlaefliGraph"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["SchlaefliGraph","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["SchlaefliGraph","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["SchlaefliGraph","EdgeList"]]

Default embedding

GraphPlot@GraphData["SchlaefliGraph","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["SchlaefliGraph","Graph","Degenerate"]//StyleGraphs

Brouwer == complement of {"GeneralizedQuadrangle", {2, 4}}

cij=Flatten[Table[{a,b},{a,5},{b,a+1,6}],1];
linesatob=Flatten[Table[If[a≠b,{a,b+6},Sequence@@{}],{a,6},{b,6}],1];
linesabtoc=Flatten[Table[If[{Mod[n,6,1]}⋂cij[[k]]≠{},{n,k+12},Sequence@@{}],{n,12},{k,15}],1];
linesctoc=Flatten[Table[If[cij[[j]]⋂cij[[k]]=={},{j+12,k+12},Sequence@@{}],{j,15},{k,15}],1];

Q4=Graph[UndirectedEdge@@@Union[Sort/@Sort[Flatten[{linesatob,linesabtoc,linesctoc},1]]]]

RecognizeGraph[Q4]

RecognizeGraph[GraphComplement[Q4]]

Wikipedia

str=Import["http://upload.wikimedia.org/wikipedia/commons/f/f0/Schläfli_graph.svg","String"]

RecognizeGraph[g8=SVGToGraph[str]]

e=ToCommonEdges[g8,GraphData["SchlaefliGraph"]]

Mean[e]

e2=(#-400{1,1})/400&/@e

ShowGraph[gg=ChangeVertices[Graph["SchlaefliGraph"],e2],Axes->True]

RecognizeGraph[gg]

Vertices[gg]

Mean[%%]

GraphData["SchlaefliGraph",{"Regular","Hamiltonian"}]

{V[newg],M[newg]}

LCFFromEmbedding[gg]

RecognizeGraph[newg =LCFNotation[{{-12, -11, -10, -7, -6, -2, 4, 5, 6, 10, 11, 12, 13, 14}, {-11, -10, -7, -6, -5, -4, -3, 3, 6, 7, 10, 11, 13, 14}, {-12, -11, -10, -4, -3, 2, 3, 4, 7, 10, 11, 12, 13, 14}}, 9]]

GraphPlot[ChangeVertices[Graph["SchlaefliGraph"],e2],Method->None,VertexLabeling->True]

GraphData["SchlaefliGraph","Degrees"]//Union

RecognizeGraph[newg =LCFNotation[{{-12, -11, -10, -7, -6, -2, 4, 5, 6, 10, 11, 12, 13, -13}, {-11, -10, -7, -6, -5, -4, -3, 3, 6, 7, 10, 11, 13,-13}, {-12, -11, -10, -4, -3, 2, 3, 4, 7, 10, 11, 12, 13, -13}}, 9]]

RecognizeGraph[newg =LCFNotation[{{-3,-4,-10,-11,-12,-13,13,12,11,10,7,4,3,2},{-3,-4,-5,-6,-7,-10,-11,-13,13,11,10,7,6,3},{-2,-6,-7,-10,-11,-12,-13,13,12,11,10,6,5,4}}, 9]]

GetGraphData["SchlaefliGraph","File"]

ShowGraph[newg]

e=Prepend[GraphData["SchlaefliGraph","Embeddings"],ToCommonEdges[newg,Graph["SchlaefliGraph"]]]

distanceregular.org

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

RecognizeGraph[g]

Complement of {"GeneralizedQuadrangle", {2, 4}}

GraphData[{"GeneralizedQuadrangle",{2,4}},"Graph","All"]

GraphComplement/@%

Bilateral

GraphData["SchlaefliGraph","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["SchlaefliGraph","Graph","Circular"]//StyleGraphs

Planar

GraphData["SchlaefliGraph","Planar"]

GraphData["SchlaefliGraph","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["SchlaefliGraph",{"RCN","CN"}]

GraphData["SchlaefliGraph","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["SchlaefliGraph"],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["SchlaefliGraph"],TimeConstraint->60])//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["SchlaefliGraph"],"CrossingNumber"->cn]]

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["SchlaefliGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["SchlaefliGraph","V"]&]]//Timing

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["SchlaefliGraph","Circulant"]

GraphData["SchlaefliGraph","Graph","Circulant"]//StyleGraphs

Halin

GraphData["SchlaefliGraph","Halin"]

GraphData["SchlaefliGraph","Graph","Halin"]//StyleGraphs

LCF

GraphData["SchlaefliGraph","LCF"]

GraphData["SchlaefliGraph","HamiltonianCycleCount"]

GraphData["SchlaefliGraph","LCFSignature"]

GraphData["SchlaefliGraph","LCFNotations"]

GraphData["SchlaefliGraph","Graph","LCF"]//StyleGraphs

Tally[LCFNotation[#][[2]]&/@%]

LCF compute random

FindRandomLCFNotation[GraphData["SchlaefliGraph"],TimeConstraint->3600]

FindRandomLCFNotation[GraphData["SchlaefliGraph"],TimeConstraint->10 3600]

LCF compute 10^5

Tally[(lcf=LCF[GraphData["SchlaefliGraph"],10^5])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

LCF compute 10^6

Tally[(lcf=LCF[GraphData["SchlaefliGraph"],10^6])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

LCF compute 10^7

Tally[(lcf=LCF[GraphData["SchlaefliGraph"],10^7])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

LCF compute 5*^7

Tally[(lcf=LCF[GraphData["SchlaefliGraph"],5*^7])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

UnitDistance

GraphData["SchlaefliGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["SchlaefliGraph"],Debug->True]//Timing

GraphData["SchlaefliGraph","Graph","UnitDistance"]//StyleGraphs

Matchstick

GraphData["SchlaefliGraph","Matchstick"]

GraphData["SchlaefliGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["SchlaefliGraph","Edges"]]

3D All

GraphData["SchlaefliGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["SchlaefliGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["SchlaefliGraph","Polyhedral"]

GraphData["SchlaefliGraph","Graph",{"3D","Polyhedron"}]

GraphData

GraphClassString

GraphClassString["SchlaefliGraph"]

GraphDataString

GraphDataString[SchlaefliGraph,
"Name"->"Schläfli graph",
"StandardName"->"SchlaefliGraph",
"AlternateStandardNames"->{"SchlafliGraph"},
"Information"->"SchlaefliGraph",
"Vertices"->None,
"Skip"->{"Perfect"},
TimeConstraint->60,
Debug->True
]