Truncated Pentakis Dodecahedral Graph

Author

Eric W. Weisstein
September 2, 2022

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

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

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

Graphic

GraphicsRow[Take[GraphData["TruncatedPentakisDodecahedralGraph","Graph","All"],2]//StyleGraphs[#,VertexSize->{"Scaled",.012}]&,ImageSize->500]

Embeddings

<<MathWorld`Graphs`

GraphData["TruncatedPentakisDodecahedralGraph","StandardName"]

Primary

GraphData["TruncatedPentakisDodecahedralGraph"]//StyleGraphs[#,ImageSize->Small,VertexSize->{"Scaled",.01}]&

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["TruncatedPentakisDodecahedralGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

GraphData["TruncatedPentakisDodecahedralGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

g=RotateGraph[MinimalCrossingGraph[GraphData["TruncatedPentakisDodecahedralGraph"],"CrossingNumber"->-1]]

RecognizeGraph[g]

Tutte embeddings

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

Tally[ReverseSort[Length/@faces]]

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

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{,RotateGraph[,-2Pi/6],}

Tutte embedding construction 6 #1

GraphPlot[RotateGraph[]]

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

RecognizeGraph[g]

Tutte embedding construction 6 #2

GraphPlot[RotateGraph[,-2Pi/6]]

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

RecognizeGraph[g]

Tutte embedding construction 5

GraphPlot[RotateGraph[]]

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

RecognizeGraph[g]

Halin

GraphData["TruncatedPentakisDodecahedralGraph","Halin"]

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

LCF

GraphData["TruncatedPentakisDodecahedralGraph","LCF"]

GraphData["TruncatedPentakisDodecahedralGraph","HamiltonianCycleCount"]

GraphData["TruncatedPentakisDodecahedralGraph","LCFSignature"]

GraphData["TruncatedPentakisDodecahedralGraph","LCFNotations"]

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

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

LCF compute random

ComputeRandomLCFNotation["TruncatedPentakisDodecahedralGraph",2,TimeConstraint->120]

lcf={{{-59,59,-20,55,51,-24,47,-28,44,9,-32,5,-36,-39,-43,9,-5,5,-9,32,12,8,-5,5,-9,-55,-59,59,-5,-8,55,43,-12,17,39,13,36,9,32,28,5,24,20,16,13,-5,-9,9,-13,5,-17,-32,-44,-47,-5,-51,-9,-13,-55,-16},3}};

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

PropertyReplace[l,"LCFNotations","13.3"]

RenameGraphFiles[]

LCFGraph@@@lcf

BilateralLCFGraph/@%

LCF compute 10^5

GraphData["TruncatedPentakisDodecahedralGraph","HamiltonianCycleCount"]

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

MaxMemoryUsed[]/10.^9 "GB"

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

gs=LCFGraph@@@DeleteCases[lcf,{_,1|2}]//StyleGraphs

BilateralLCFGraph/@gs//Timing

BilateralLCFGraph/@%

lcf=LCFNotation/@{,,,}

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

PropertyReplace[l,"LCFNotations","13.3"]

RenameGraphFiles[]

LCF compute 10^6

GraphData["TruncatedPentakisDodecahedralGraph","HamiltonianCycleCount"]

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1|2}]//StyleGraphs

<<MathWorld`Graphs`

gs3=BilateralLCFGraph[Sequence@@#,"ReturnGraph"->True]&/@Cases[lcf,{_,3}]

Length[gs3]

Select[gs3,BilateralGraphEmbeddingQ]//Timing

(gs2=BilateralLCFGraph[Sequence@@#,"ReturnGraph"->True]&/@Cases[lcf,{_,2}])//Timing

Select[gs2,BilateralGraphEmbeddingQ]//Timing

LCF compute 10^7

GraphData["TruncatedPentakisDodecahedralGraph","HamiltonianCycleCount"]

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

UnitDistance

GraphData["TruncatedPentakisDodecahedralGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

3600

With[{gn="TruncatedPentakisDodecahedralGraph",g=},
Print[{gn,FindSubgraphUnitDistanceEmbedding[g,"Results"->All,TimeConstraint->3600,Debug->Full]//Timing}]]

gs=UniqueEmbeddings[l]

Matchstick

GraphData["TruncatedPentakisDodecahedralGraph","Matchstick"]

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

3D default

Graph3D[GraphData["TruncatedPentakisDodecahedralGraph","Edges"]]

3D All

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

3D UnitDistance construction

UnitDistanceEmbeddingRefine@Graph3D[GraphData["TruncatedPentakisDodecahedralGraph","Edges"]]

3D UnitDistance

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

3D Polyhedron Construction

g=PolyhedronGraph[PolyhedronData["TruncatedPentakisDodecahedron","Polyhedron"]/.Polyhedron[v_,f_]:>Polyhedron[v/PolyhedronData["TruncatedPentakisDodecahedron","Midradius"],f]]

3D Polyhedron

GraphData["TruncatedPentakisDodecahedralGraph","Polyhedral"]

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

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

3D Canonical

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

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

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData["TruncatedPentakisDodecahedralGraph"],-1]}]

Graphics3D[{FaceForm[Green,Red],poly=PolyhedronData["TruncatedPentakisDodecahedron","Polyhedron"]/.Polyhedron[v_,f_]:>Polyhedron[v/PolyhedronData["TruncatedPentakisDodecahedron","Midradius"],f]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->100,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->500,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

SimplePolyhedronQ[can]

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

g=PolyhedronGraph[can]

Properties

GraphData["TruncatedPentakisDodecahedralGraph",{"V","E","F"}]

PolyhedralEmbeddings

GraphData["TruncatedPentakisDodecahedralGraph","PolyhedralEmbeddings"]

GraphData

GraphDataClassString

GraphDataClassString["TruncatedPentakisDodecahedralGraph"]

GraphDataString

file=OpenWrite["~/tpd.m"]

WriteString[file,GraphDataString[
g,
"StandardName"->"TruncatedPentakisDodecahedralGraph",
"Name"->"truncated pentakis dodecahedral graph",
"Information"->"TruncatedPentakisDodecahedralGraph",
"Embeddings"->None
]]

Close[file]