Schläfli Double Sixes Graph

Author

Eric W. Weisstein
August 25, 2022

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

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

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

Classes

GraphData["SchlaefliDoubleSixesGraph","Classes"]

Embeddings

GraphData["SchlaefliDoubleSixesGraph","StandardName"]

Primary

GraphData["SchlaefliDoubleSixesGraph"]//StyleGraphs

Default embedding

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

Graph[EdgeList[g]]

GraphicsToGraph@GraphPlot[GraphData["SchlaefliDoubleSixesGraph","Edges"]]

(Sqrt[2])/2.

square=CirclePoints[1,4];
square3=Flatten[{#1+(#2-#1)/3,#1+(#2-#1)2/3}&@@@Partition[square,2,1,1],1];

v=Join[CirclePoints[(1/(3 (Sqrt[2]))),6],square,square3,CirclePoints[.8,6],CirclePoints[(2^(-1/2)),4]];
g=Graph[Range[Length[v]],UndirectedEdge@@@Join[
{{18,7},{7,11},{12,8},{8,13},{14,9},{9,15},{16,10},{10,17},{17,5},{5,14},{18,2},{2,13},{16,4},{4,26},{15,6},{6,25}}
],VertexCoordinates->v,VertexLabels->Automatic]

All

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

Construction as a configuration

MatrixForm[conf={{1,4,26,20,21},{1,5,25,10,8},{2,4,29,14,18},{2,6,25,11,9},{3,5,29,13,17},{3,6,26,24,23},{7,10,27,22,23},{7,11,30,15,17},{8,12,30,14,16},{9,12,27,20,19},{16,13,28,22,21},{18,15,28,24,19}}]

(g=MyIncidenceGraph[conf])//RecognizeGraph

Degenerate

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

Planar

GraphData["SchlaefliDoubleSixesGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

CrossingNumber[GraphData["SchlaefliDoubleSixesGraph"],TimeConstraint->10]//AbsoluteTiming

CrossingNumber[GraphData["SchlaefliDoubleSixesGraph"],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData["SchlaefliDoubleSixesGraph"],"CrossingNumber"->12]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

Halin

GraphData["SchlaefliDoubleSixesGraph","Halin"]

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

LCF

GraphData["SchlaefliDoubleSixesGraph","LCF"]

GraphData["SchlaefliDoubleSixesGraph","LCFSignature"]

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

UnitDistance

GraphData["SchlaefliDoubleSixesGraph","UnitDistance"]

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

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

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData["SchlaefliDoubleSixesGraph","Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData["SchlaefliDoubleSixesGraph"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{"SchlaefliDoubleSixesGraph",l=FindSubgraphUnitDistanceEmbedding[GraphData["SchlaefliDoubleSixesGraph"],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData["SchlaefliDoubleSixesGraph","Matchstick"]

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

3D default

Graph3D[GraphData["SchlaefliDoubleSixesGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["SchlaefliDoubleSixesGraph","Polyhedral"]

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

GraphData

GraphClassString

GraphClassString["SchlaefliDoubleSixesGraph"]

GraphDataString

Print[GraphDataString[g,
"Name"->"Schläfli double-sixes graph",
"StandardName"->{"SchlaefliDoubleSixesGraph"},
"AlternateStandardName"->{"SchlafliDoubleSixesGraph"},
"NotationRules"->{"Configuration"->{{1,4,26,20,21},{1,5,25,10,8},{2,4,29,14,18},{2,6,25,11,9},{3,5,29,13,17},{3,6,26,24,23},{7,10,27,22,23},{7,11,30,15,17},{8,12,30,14,16},{9,12,27,20,19},{16,13,28,22,21},{18,15,28,24,19}}},
TimeConstraint->120,
Debug->True
]]