Pentagonal Icositetrahedral Graph

Author

Eric W. Weisstein
August 23, 2022

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

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

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

Graph

Primary

GraphData["PentagonalIcositetrahedralGraph"]//StyleGraphs

All

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

Degenerate

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

LCF

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

MinimalCrossing

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

Perspective

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

Planar

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

UnitDistance

GraphData["PentagonalIcositetrahedralGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

Matchstick

GraphData["PentagonalIcositetrahedralGraph","Matchstick"]

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

MinimalCrossing

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

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

Default embedding

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

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

RecognizeGraph[g]

Tutte embeddings

faces=IGFaces[GraphData["PentagonalIcositetrahedralGraph"]]

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

3D default

Graph3D[GraphData["PentagonalIcositetrahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["PentagonalIcositetrahedralGraph","Polyhedral"]

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

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

3D Canonical

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

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

3D Canonical construction

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

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"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]

Embeddings

GraphDataEmbeddings[GraphData["PentagonalIcositetrahedralGraph","EdgeList"]]

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["PentagonalIcositetrahedralGraph"]]]

Properties

GraphData[g,"Missing"]

GraphData[g,{"V","M"}]

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

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["PentagonalIcositetrahedron","Polyhedron"],#,"Simplification"->RootReduce]&/@{{(NumericalApproximation[Root, (RowDefault["0.544", FEExpressionsNumericalApproximationElider]), Root[-1+#1+((#1)^2)+((#1)^3)&,1], ]),0,1},{(NumericalApproximation[Root, (RowDefault["0.296", FEExpressionsNumericalApproximationElider]), Root[-1+3 #1+((#1)^2)+((#1)^3)&,1], ]),0,1},{0,0,1},{1,0,1},{1,1,1}}

GraphData["PentagonalIcositetrahedralGraph","Graph","Perspective"]

Embeddings by projection

PolyhedronData[poly]

Degenerate

GraphPlot[SkeletonGraph[PolyhedronData["PentagonalIcositetrahedron","Polyhedron"],#],Method->None]&/@{1.9,2.7,3,3.5,5,6}

rs=RecognizeGraph/@(gs=ToCombinatoricaGraph/@{,})

ToCommonEdges[#,rs[[1]]]&/@gs

Construction

Rectangular

GraphData["PentagonalIcositetrahedralGraph"]

GraphEmbedding[%]

Skelteton graph 1.9

SkeletonGraph[PolyhedronData["PentagonalIcositetrahedron","Polyhedron"],1.9]

ScaleGraph[RotateGraph[%,{20,32}]]

Skelteton graph 3.5

SkeletonGraph[PolyhedronData["PentagonalIcositetrahedron","Polyhedron"],3.5]

GraphPlot@ScaleGraph[RotateGraph[%,{22,29}]]

Edge list

GraphPlot[GraphData["PentagonalIcositetrahedralGraph","EdgeList"]]

GraphData

GraphClasses

GraphClassString[g]