Deltoidal Icositetrahedral Graph

Author

Eric W. Weisstein
August 22, 2022

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

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

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

Graph

Default

Embedding by Ed Pegg, Jr.
May 2007

GraphData["DeltoidalIcositetrahedralGraph"]//StyleGraphs

All

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

Degenerate

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

LCF

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

GraphData["DeltoidalIcositetrahedralGraph","LCF"]

MinimalCrossing

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

Perspective

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

Planar

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

UnitDistance

GraphData["DeltoidalIcositetrahedralGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

gs={,,,};

Matchstick

GraphData["DeltoidalIcositetrahedralGraph","Matchstick"]

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

MinimalCrossing

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

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

Default embedding

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

ee=Sort[Sort/@{e[21,24,27,26,25,16,7,4,1,2,3,12,21],e[2,5,8,17,26,23,20,11,2],e[3,6,9,18,27],e[7,8,9],e[16,17,18,15,12,11,10,13,16],e[21,20,19,22,25],e[1,10,19],e[4,5,6,15,24],e[4,13,22,23,24]}];

q=-40Degree;
v=Flatten[Table[{i,j}+k Through[{Cos,Sin}[q]],{i,0,2},{j,0,2},{k,0,2}],2];
g=IndexGraph@VertexDelete[Graph[Range[Length[v]],UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->"Name"],14]

RecognizeGraph[g]

UnitDistanceGraphEmbeddingQ[g]

q=-45Degree;
v=Flatten[Table[{i,j}+k Through[{Cos,Sin}[q]],{i,0,2},{j,0,2},{k,0,2}],2];
g=IndexGraph@VertexDelete[Graph[Range[Length[v]],UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->"Name"],14]

Manipulate[Module[{v},
(
v=Flatten[Table[{i,j}+k Through[{Cos,Sin}[q Degree]],{i,0,2},{j,0,2},{k,0,2}],2];
IndexGraph@VertexDelete[Graph[Range[Length[v]],UndirectedEdge@@@ee,VertexCoordinates->v,VertexLabels->"Name"],14]
)
],{q,0,90}]

gs={,}

MinimalCrossing construction

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

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData["DeltoidalIcositetrahedralGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["DeltoidalIcositetrahedralGraph","Polyhedral"]

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

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

3D Canonical

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

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

3D Canonical construction

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

(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]

Embeddings

GraphDataEmbeddings[GraphData["DeltoidalIcositetrahedralGraph","EdgeList"]]

Perspective projections

PolyhedronProjectionGraph[PolyhedronData["DeltoidalIcositetrahedron","Polyhedron"],#,"Simplification"->FullSimplify]&/@{{0,1,0},{1,1,0},{1,1+(Sqrt[2]),0},{1,1,1,π/2},{1,(1/17) (6+(Sqrt[2])),1,-ArcTan[(Sqrt[(5/82)+(3/(82 (Sqrt[2])))])]},{1,2(Sqrt[2]),1,-ArcTan[(2/(Sqrt[5]))]},{1,2,1,-ArcTan[(Sqrt[(2/3)])]}}

GraphData["DeltoidalIcositetrahedralGraph","Graph","Perspective"]

Properties

Classes

GraphData[g,"Classes"]

Matrices

GraphDataMatrices[g]

Properties

GraphData[g,"Missing"]

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

GraphDataTable[g]//TableForm//TraditionalForm

Embeddings by Projection

PolyhedronData[poly]

Degenerate

GraphPlot[g=SkeletonGraph[PolyhedronData[poly,"Faces"]//Normal,2],Method->None]

r=RecognizeGraph[ToCombinatoricaGraph[g]]

ToCommonEdges[g,r]//N

GraphPlot[g]

GraphPlot3D[g]

GraphData

GraphClasses

GraphClassString[g]