Small Rhombicosidodecahedral Graph

Author

Eric W. Weisstein
November 3, 2023

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

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

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

Embeddings

<<MathWorld`Graphs`

Names

GraphData["SmallRhombicosidodecahedralGraph","StandardName"]

GraphData["SmallRhombicosidodecahedralGraph","StandardNames"]

Primary

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

All

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

Construct from Read and Wilson

g=Graph[Range[60],UndirectedEdge@@@{{1,2},{1,3},{1,5},{1,32},{2,3},{2,4},{2,8},{3,7},{3,24},{4,6},{4,7},{4,9},{5,8},{5,13},{5,55},{6,10},{6,11},{6,14},{7,9},{7,12},{8,10},{8,13},{9,11},{9,15},{10,16},{10,17},{11,14},{11,18},{12,15},{12,23},{12,28},{13,16},{13,48},{14,17},{14,19},{15,20},{15,23},{16,17},{16,29},{17,21},{18,19},{18,20},{18,22},{19,22},{19,25},{20,26},{20,27},{21,25},{21,29},{21,37},{22,26},{22,30},{23,27},{23,31},{24,28},{24,32},{24,40},{25,30},{25,33},{26,27},{26,34},{27,35},{28,31},{28,36},{29,37},{29,45},{30,33},{30,38},{31,36},{31,39},{32,40},{32,60},{33,37},{33,41},{34,35},{34,38},{34,42},{35,39},{35,42},{36,40},{36,49},{37,44},{38,41},{38,43},{39,47},{39,49},{40,58},{41,43},{41,46},{42,43},{42,47},{43,50},{44,45},{44,46},{44,51},{45,48},{45,51},{46,50},{46,56},{47,49},{47,52},{48,53},{48,55},{49,54},{50,52},{50,56},{51,53},{51,56},{52,54},{52,57},{53,55},{53,59},{54,57},{54,58},{55,60},{56,57},{57,59},{58,59},{58,60},{59,60}},VertexCoordinates->(10^(-4)){{8244,5087},{10106,3018},{10966,5666},{11825,3018},{12177,-7023},{12177,-329},{12357,4655},{12468,-4254},{13507,3376},{13859,-3243},{14039,1740},{14608,7156},{14719,-5890},{14899,-907},{14899,4387},{15250,-4254},{15430,-2543},{15759,1740},{16290,104},{16290,3376},{17150,-2543},{17150,1288},{17150,6023},{17150,10161},{17681,-907},{17681,2924},{17681,4387},{17681,8524},{18541,-4954},{18541,277},{18541,7035},{18541,12571},{19401,-907},{19401,2924},{19401,4387},{19401,8524},{19932,-2543},{19932,1288},{19932,6023},{19932,10161},{20792,104},{20792,3376},{21324,1740},{21652,-2543},{21832,-4254},{22183,-907},{22183,4387},{22363,-5890},{22474,7156},{23043,1740},{23223,-3243},{23575,3376},{24614,-4254},{24725,4655},{24905,-7023},{24905,-329},{25257,3018},{26117,5666},{26976,3018},{28838,5087}}]

RecognizeGraph@g

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["SmallRhombicosidodecahedralGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

DegenerateGraphEmbeddingTypes/@GraphData["SmallRhombicosidodecahedralGraph","Graph","Degenerate"]

Perspective

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

Perspective construction

GraphData["SmallRhombicosidodecahedralGraph","Polyhedron"]

Bilateral

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

Circular

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

Planar

GraphData["SmallRhombicosidodecahedralGraph","Graph","Planar"]//StyleGraphs[#,VertexSize->{"Scaled",.02}]&

Tutte embeddings

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

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData["SmallRhombicosidodecahedralGraph"],"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 4

GraphPlot@

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

RecognizeGraph[g]

Tutte embedding construction 3

GraphPlot@

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

RecognizeGraph[g]

MinimalCrossing

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

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

LCF

GraphData["SmallRhombicosidodecahedralGraph","LCF"]

GraphData["SmallRhombicosidodecahedralGraph","HamiltonianCycleCount"]

GraphData["SmallRhombicosidodecahedralGraph","LCFSignature"]

GraphData["SmallRhombicosidodecahedralGraph","LCFNotations"]

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

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

LCF compute random

FindRandomLCFNotation[GraphData["SmallRhombicosidodecahedralGraph"],"MinOrder"->3,TimeConstraint->120]//Timing//AbsoluteTiming

FindRandomLCFNotation[GraphData["SmallRhombicosidodecahedralGraph"],"MinOrder"->3,TimeConstraint->3600]//Timing//AbsoluteTiming

FindRandomLCFNotation[GraphData["SmallRhombicosidodecahedralGraph"],"MinOrder"->3,TimeConstraint->5 3600]//Timing//AbsoluteTiming

LCF compute 10^5

Tally[(lcf=LCF[GraphData["SmallRhombicosidodecahedralGraph"],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]

LCF compute 10^6

Tally[(lcf=LCF[GraphData["SmallRhombicosidodecahedralGraph"],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]

LCF compute 10^7

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1|2}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,3])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

Tally[(LCFNotation/@gs)[[All,-1]]]

lcf={{{{-20,20},{-6,18},{-8,17},{14,15},{-11,2},{6,12},{-2,4},{-15,-14},{-18,-17},{-20,20},{-4,18},{-6,17},{4,15},{11,14},{2,8},{3,6},{-4,-2},{-14,-12},{-15,-3},{-18,-17}},3},{{{-20,20},{12,18},{-4,10},{-6,9},{6,7},{-17,-9},{-18,4},{-20,20},{17,18},{6,7},{-6,-4},{-7,4},{-10,-9},{-12,6},{3,4},{-6,-4},{-7,9},{-3,6},{-4,4},{-18,-6}},3},{{{-20,20},{17,18},{3,15},{12,14},{2,4},{-6,-3},{-8,-2},{-14,-11},{-15,-4},{-17,6},{-18,4},{-20,20},{17,18},{14,15},{-4,2},{-12,-6},{-2,11},{-15,-14},{-17,8},{-18,6}},3}};

l={{"SmallRhombicosidodecahedralGraph",{0,Missing["NotAvailable",lcf]}}}

PropertyReplace[l,"LCFNotations","14.1"]

RenameGraphFiles[]

LCF compute 2*^7

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1|2}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,3])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

Tally[(LCFNotation/@gs)[[All,-1]]]

lcf={{{{-20,20},{-6,18},{-8,17},{14,15},{-11,2},{6,12},{-2,4},{-15,-14},{-18,-17},{-20,20},{-4,18},{-6,17},{4,15},{11,14},{2,8},{3,6},{-4,-2},{-14,-12},{-15,-3},{-18,-17}},3},{{{-20,20},{-4,18},{-6,17},{-9,15},{-12,14},{11,12},{-15,9},{6,8},{-18,-17},{-20,20},{3,18},{15,17},{2,12},{-6,-3},{-2,9},{-9,-8},{-11,6},{-12,4},{-15,-14},{-18,-17}},3},{{{-20,20},{-4,18},{-6,17},{9,15},{-9,7},{4,6},{-12,-11},{-18,-17},{-20,20},{-4,18},{4,17},{-7,-6},{-9,6},{3,4},{-4,12},{9,11},{-3,6},{-4,4},{-15,-6},{-18,-17}},3},{{{-20,20},{-4,18},{-6,17},{11,15},{8,9},{-9,2},{3,6},{-12,-2},{-17,-15},{-18,-3},{-20,20},{17,18},{-8,-6},{-9,15},{-11,4},{2,12},{6,9},{-2,4},{-15,-4},{-18,-17}},3},{{{-20,20},{-4,18},{9,17},{-7,-6},{-9,6},{-10,4},{-18,-12},{-20,20},{6,18},{-4,4},{-6,3},{-9,7},{4,6},{-4,-3},{-6,12},{9,10},{-4,7},{4,6},{-7,-6},{-18,-17}},3},{{{-20,20},{3,18},{15,17},{2,12},{-6,-3},{-2,9},{-9,-8},{-15,-11},{-17,6},{-18,4},{-20,20},{17,18},{4,15},{-4,2},{-9,-6},{-12,-2},{-4,11},{-15,9},{6,8},{-18,-17}},3},{{{-20,20},{12,18},{-4,10},{-6,9},{6,7},{-17,-9},{-18,4},{-20,20},{17,18},{6,7},{-6,-4},{-7,4},{-10,-9},{-12,6},{3,4},{-6,-4},{-7,9},{-3,6},{-4,4},{-18,-6}},3},{{{-20,20},{17,18},{3,15},{12,14},{2,4},{-6,-3},{-8,-2},{-14,-11},{-15,-4},{-17,6},{-18,4},{-20,20},{17,18},{14,15},{-4,2},{-12,-6},{-2,11},{-15,-14},{-17,8},{-18,6}},3}};

l={{"SmallRhombicosidodecahedralGraph",{0,Missing["NotAvailable",lcf]}}}

PropertyReplace[l,"LCFNotations","14.1"]

RenameGraphFiles[]

UnitDistance

GraphData["SmallRhombicosidodecahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["SmallRhombicosidodecahedralGraph","VertexList"],GraphData["SmallRhombicosidodecahedralGraph","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["SmallRhombicosidodecahedralGraph","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

FindUnitDistanceEmbedding[GraphData["SmallRhombicosidodecahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["SmallRhombicosidodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["SmallRhombicosidodecahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["SmallRhombicosidodecahedralGraph","Polyhedral"]

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

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

3D Canonical

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

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

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],#,"Simplification"->FullSimplify]&/@{{0,0,1},{1,(1/6) (3-(Sqrt[5])),(1/6) (5-(Sqrt[5])),ArcTan[(Sqrt[(1/2)+(1/(Sqrt[5]))])]},{1,(1/6) (1+(Sqrt[5])),(1/6) (-1+(Sqrt[5]))},{(1/22) (-5+7 (Sqrt[5])),(1/22) (1+3 (Sqrt[5])),1,-ArcTan[(Sqrt[(5/2)-(Sqrt[5])])]},{0,(1/10) (5+(Sqrt[5])),1},{0,1/2,1},{0,(1/6) (5-(Sqrt[5])),1},{0,1,1},{1,1,1,-ArcCot[(Sqrt[3 (9+4 (Sqrt[5]))])]},{0,(1/2)(1+(Sqrt[5])),1}}

GraphData["SmallRhombicosidodecahedralGraph","Graph","Perspective"]

Embedding as circle intersections

Nice embedding and original circular embedding code by Ed Pegg Jr.
September 23–24, 2002

InnerArc[Circle[{x1_,y1_},r1_],Circle[{x2_,y2_},r2_]]:=Module[
{
g=Coordinates/@Intersections[Circle[{x1,y1},r1],Circle[{x2,y2},r2]],
m,n
},
m=Mod[ArcTan[g[[2,1]] -x1,g[[2,2]] -y1],2 Pi];n =Mod[ArcTan[g[[1,1]] -x1,g[[1,2]]-y1],2 Pi];
If[Abs[m-n]<Pi,
If[m>n,Circle[{x1,y1},r1,{n,m}],Circle[{x1,y1},r1,{m,n}]],
If[m>n,Circle[{x1,y1},r1,{m,n+2 Pi}],Circle[{x1,y1},r1,{n,m+2 Pi}]]
]]

pts=Select[Flatten[Intersections@@@Subsets[{Circle[{0, 0}, .8],
Circle[{0,0},2.1],
Circle[{0,0},2.34223],
Circle[{0,-2.35},1.45],
Circle[{-2.23498,-0.72619},1.45],
Circle[{-1.38129,1.90118},1.45],
Circle[{1.38129,1.901189},1.45],
Circle[{2.23498,-0.726189},1.45],
Circle[{0.,1.},0.8],
Circle[{0.951056,0.309016},0.8],
Circle[{0.58778,-0.809016},0.8],
Circle[{-0.58778,-0.809016},0.8],
Circle[{-0.95105,0.309016},0.8]},{2}]],Im[Coordinates[#]]=={0,0}&];

{
GraphData["SmallRhombicosidodecahedralGraph"]//StyleGraphs,
Graphics[{Circle[{0, 0}, .8],
Circle[{0,0},2.1],
Circle[{0,0},2.34223],
Table[Circle[{Sin[n],Cos[n]}, .8],{n,0,2 Pi,2 Pi/5}],
Table[InnerArc[Circle[2.35{Sin[Pi+n(Pi/5)],Cos[Pi+n(Pi/5)]},1.45],Circle[{0,0},2.342237752463081]],
{n,0,8,2}
],
Red,PointSize[.03],pts
},AspectRatio->Automatic]
}