Snub Dodecahedral Graph

Author

Eric W. Weisstein
June 21, 2023

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

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

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

Graph

<<MathWorld`Graphs`

Names

GraphData["SnubDodecahedralGraph","StandardName"]

GraphData["SnubDodecahedralGraph","StandardNames"]

Primary

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

Construction from Read and Wilson

Graph[Range[60],UndirectedEdge@@@{{1,2},{1,3},{1,6},{1,17},{1,32},{2,3},{2,4},{2,6},{2,12},{3,5},{3,16},{3,17},{4,7},{4,8},{4,12},{4,13},{5,7},{5,10},{5,11},{5,16},{6,12},{6,21},{6,57},{7,8},{7,9},{7,10},{8,9},{8,13},{8,14},{9,10},{9,15},{9,19},{10,11},{10,19},{11,16},{11,20},{11,23},{12,13},{12,21},{13,14},{13,22},{14,18},{14,22},{14,29},{15,18},{15,19},{15,25},{15,26},{16,17},{16,23},{17,24},{17,32},{18,25},{18,29},{18,30},{19,26},{19,27},{20,23},{20,27},{20,37},{20,38},{21,28},{21,53},{21,57},{22,28},{22,29},{22,35},{23,31},{23,38},{24,31},{24,32},{24,51},{24,56},{25,26},{25,30},{25,33},{26,27},{26,34},{27,34},{27,37},{28,35},{28,46},{28,53},{29,30},{29,35},{30,33},{30,36},{31,38},{31,42},{31,51},{32,56},{32,60},{33,36},{33,39},{33,43},{34,37},{34,39},{34,41},{35,40},{35,46},{36,40},{36,43},{36,48},{37,38},{37,41},{38,42},{39,41},{39,43},{39,44},{40,46},{40,47},{40,48},{41,44},{41,45},{42,45},{42,50},{42,51},{43,44},{43,48},{44,45},{44,49},{45,49},{45,50},{46,47},{46,53},{47,48},{47,52},{47,54},{48,52},{49,50},{49,52},{49,55},{50,51},{50,55},{51,56},{52,54},{52,55},{53,57},{53,58},{54,55},{54,58},{54,59},{55,59},{56,59},{56,60},{57,58},{57,60},{58,59},{58,60},{59,60}},VertexCoordinates->(10^(-4)){{3802,544},{6486,-4048},{6489,-2026},{8407,-4677},{8413,-1404},{8506,-14031},{9598,-3043},{10522,-6353},{10526,-4331},{11720,-1024},{11724,998},{11780,-12268},{11783,-10245},{12444,-6982},{12450,-3709},{12664,3528},{12668,5551},{13635,-5348},{13642,-1653},{13647,1620},{13701,-12896},{13707,-9624},{14585,2900},{14591,6172},{14787,-4513},{14791,-2491},{14833,-19},{14892,-11262},{15218,-9140},{15222,-7117},{15777,4534},{16204,9523},{16709,-5142},{16714,-1869},{17140,-9768},{17146,-6496},{17156,-517},{17160,1505},{17900,-3508},{18331,-8134},{19078,-1145},{19084,2127},{19318,-5533},{19322,-3510},{20269,489},{21230,-9771},{21233,-7749},{21240,-6161},{21246,-2889},{21776,-8},{21780,2014},{22431,-4527},{23151,-10399},{23157,-7127},{23698,-637},{23704,2636},{23816,-14058},{24343,-8765},{24889,997},{28573,499}}]

RecognizeGraph@g

All

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

Bilateral

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

Circular

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

Planar

GraphData["SnubDodecahedralGraph","Planar"]

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

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["SnubDodecahedralGraph"]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData["SnubDodecahedralGraph"],"OuterFace"->#]&/@faces

gs=Annotate[#,VertexLabels->None]&/@{}

Tutte embedding construction 5

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

RecognizeGraph[g]

Tutte embedding construction 3 #1

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

RecognizeGraph[g]

Tutte embedding construction 3 #2

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

RecognizeGraph[g]

MinimalCrossing

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

Degenerate

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

LCF

GraphData["SnubDodecahedralGraph","HamiltonianCycleCount"]

GraphData["SnubDodecahedralGraph","LCFNotations"]

GraphData["SnubDodecahedralGraph","LCFSignature"]

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

LCFNotation/@GraphData["SnubDodecahedralGraph","Graph","LCF"]

Tally[%[[All,-1]]]

LCF compute 10^5

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

(gs=DeleteMissing[BilateralLCFGraph@@@DeleteCases[lcf,{_,1}]])//Timing

Length[gs]

l={{"SnubDodecahedralGraph",{0,Missing["NotAvailable",LCFNotation/@{,,}]}}}

PropertyReplace[l,"LCFNotations","13.4"]

RenameGraphFiles[]

LCF compute 10^6

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

(gs=DeleteMissing[BilateralLCFGraph@@@DeleteCases[lcf,{_,1}]])//Timing

Length[gs]

Perspective

GraphData["SnubDodecahedralGraph","Graph","Perspective"]//StyleGraphs

UnitDistance

GraphData["SnubDodecahedralGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

Matchstick

GraphData["SnubDodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["SnubDodecahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["SnubDodecahedralGraph","Polyhedral"]

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

Graphics3D[{FaceForm[Green,Red],GraphData["SnubDodecahedralGraph","Polyhedron"]}]

3D Canonical

GraphData["SnubDodecahedralGraph","Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData["SnubDodecahedralGraph","Polyhedron","Canonical"],Yellow,Sphere[]}]

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphDataMatrices[g]

Properties

TableForm[GraphDataTable[g,x],TableDepth->2]//TraditionalForm

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["SnubDodecahedron","Polyhedron"],#,"Simplification"->RootReduce]&/@{{(NumericalApproximation[Root, (RowDefault["0.372", FEExpressionsNumericalApproximationElider]), Root[2627641-31451802 ((#1)^2)+102899511 ((#1)^4)-94533804 ((#1)^6)+15347799 ((#1)^8)-746010 ((#1)^10)+729 ((#1)^12)&,5], ]),1,(NumericalApproximation[Root, (RowDefault["0.402", FEExpressionsNumericalApproximationElider]), Root[4096-113664 ((#1)^2)+1271808 ((#1)^4)-5318784 ((#1)^6)+5360256 ((#1)^8)-1395792 ((#1)^10)+729 ((#1)^12)&,5], ]),ArcTan[(NumericalApproximation[Root, (RowDefault["0.192", FEExpressionsNumericalApproximationElider]), Root[1289-49527 ((#1)^2)+462331 ((#1)^4)-1941444 ((#1)^6)+4223588 ((#1)^8)-4681549 ((#1)^10)+2105401 ((#1)^12)&,5], ])]},{(1/(Sqrt[3])),-1,-(NumericalApproximation[Root, (RowDefault["0.221", FEExpressionsNumericalApproximationElider]), Root[1-21 ((#1)^2)+9 ((#1)^4)&,3], ]),-ArcCot[(Sqrt[13+(22/(Sqrt[5]))])]},{1,(Sqrt[(3/5)]),-(2/(Sqrt[5])),-ArcTan[(1/2) (-7+3 (Sqrt[5]))]}}

GraphData["SnubDodecahedralGraph","Graph","Perspective"]

GraphData

GraphClassString

GraphClassString[g]