Rhombicosahedral Graph

Author

Eric W. Weisstein
September 5, 2023

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

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

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

Graphics

GraphicsRow[GraphData["RhombicosahedralGraph","Graph","All"]//StyleGraphs,ImageSize->200]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["RhombicosahedralGraph","StandardName"]

GraphData["RhombicosahedralGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["RhombicosahedralGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Perspective

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

Perspective construction

GraphData["RhombicosahedralGraph","Graph","3D"]

Manipulate[PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{1,0,1,-Pi/2}]

PolyhedronProjectionGraph[GraphData["RhombicosahedralGraph","Polyhedron"],{1,1,1}]

Bilateral

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

Circular

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

Planar

GraphData["RhombicosahedralGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData["RhombicosahedralGraph"],TimeConstraint->1])//AbsoluteTiming

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

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

(cn=CrossingNumber[GraphData["RhombicosahedralGraph"],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["RhombicosahedralGraph"],TimeConstraint->5 3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["RhombicosahedralGraph"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

Table[MinimalCrossingGraph[GraphData["RhombicosahedralGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["RhombicosahedralGraph","Circulant"]

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

Line graph

GraphData["RhombicosahedralGraph","Line"]

LineGraphQ[GraphData["RhombicosahedralGraph"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "RhombicosahedralGraph"&]

RootGraph[GraphData["RhombicosahedralGraph"]]

Halin

GraphData["RhombicosahedralGraph","Halin"]

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

Rigid

GraphData["RhombicosahedralGraph","Rigid"]

GraphData["RhombicosahedralGraph","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["RhombicosahedralGraph","Flexible"]

GraphData["RhombicosahedralGraph","Graph","Flexible"]//StyleGraphs

LCF

GraphData["RhombicosahedralGraph","LCF"]

GraphData["RhombicosahedralGraph","HamiltonianCycleCount"]

GraphData["RhombicosahedralGraph","LCFSignature"]

GraphData["RhombicosahedralGraph","LCFNotations"]

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

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

LCF compute random

FindRandomLCFNotation[GraphData["RhombicosahedralGraph"],TimeConstraint->120]

LCF compute 10^5

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

LCF compute 10^6

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1|2}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,2}]]}])//Timing

Length[gs]

LCF compute 10^7

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1|2}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,2}]]}])//Timing

Length[gs]

LCF compute all

Length[FindHamiltonianCycle[GraphData["RhombicosahedralGraph"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["RhombicosahedralGraph"],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["RhombicosahedralGraph",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData["RhombicosahedralGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

GraphData["RhombicosahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["RhombicosahedralGraph","Edges"]]

3D All

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

3D UnitDistance construction from uniform polyhedron

emb=GraphData["RhombicosahedralGraph","PolyhedralEmbeddings"]

PolyhedronGraph[PolyhedronData[#,"Polyhedron"]]&/@emb

3D UnitDistance

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

3D Polyhedron

GraphData["RhombicosahedralGraph","Polyhedral"]

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

Properties

PolyhedralEmbeddings

GraphData["RhombicosahedralGraph","PolyhedralEmbeddings"]

GraphData

GraphDataClassString

GraphDataClassString["RhombicosahedralGraph"]

GraphDataString

file=OpenAppend["~/polygraphs.m"]

g=PolyhedronGraph[PolyhedronData["Rhombicosahedron","Polyhedron"]]

PolyhedronNameToGraphName[s_]:=StringReplace[s,"dron"->"dralGraph"]

lowercasedecamel[s_]:=StringTrim[StringReplace[s,RegularExpression["([A-Z])"]:>" "<>ToLowerCase["$1"]]]

With[{stdname="Rhombicosahedron",altnames={"Icosidodecadodecahedron","Rhombidodecadodecahedron"}},
WriteString[file,GraphDataString[g,
"StandardName"->PolyhedronNameToGraphName@stdname,
"AlternateStandatdNames"->PolyhedronNameToGraphName/@altnames,
"Name"->lowercasedecamel@stdname,
"AlternateNames"->lowercasedecamel/@PolyhedronNameToGraphName/@altnames,
"Information"->stdname,
"Classes"->{"UniformSkeleton"},
"Skip"->{"ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","TuttePolynomial"},
TimeConstraint->10
]]]

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