Pentatope Graph

Author

Eric W. Weisstein
January 16, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/PentatopeGraph.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Figure

GraphicsRow[StyleGraphs@GraphData["PentatopeGraph","Graph","All"],Dividers->All]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["PentatopeGraph","StandardName"]

GraphData["PentatopeGraph","StandardNames"]

HouseOfGraphs

GraphData["PentatopeGraph","HouseOfGraphsID"]

hogid=GraphData["PentatopeGraph","NotationRules","HouseOfGraphs"]

HouseOfGraphsLookup[hogid]

HouseOfGraphsGraph[hogid]//StyleGraphs[#,ImageSize->Small]&

Wikidata

GraphData["PentatopeGraph","WikidataID"]

GraphData["PentatopeGraph","WikidataURL"]

Primary

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

Read and Wilson construction

All

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

CurvedEdge

GraphData["PentatopeGraph","Graph","CurvedEdge"]//StyleGraphs

CurvedEdge construction

https://doi.org/10.1080/00029890.2025.2533060

Import["~/Unknown.jpeg"]

v=CirclePoints[5];
g=CurvedEdgeAnnotate[
Graph[Range@Length@v,UndirectedEdge@@@Subsets[Range[5],{2}],VertexCoordinates->v,VertexLabels->Automatic],
{{{1,3},1/2,-2},{{3,5},1/2,-2}}]

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["PentatopeGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

GraphData["PentatopeGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["PentatopeGraph","Circulant"]

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

Line graph

GraphData["PentatopeGraph","Line"]

LineGraphQ[GraphData["PentatopeGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["PentatopeGraph"]]

Halin

GraphData["PentatopeGraph","Halin"]

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

Rigid

GraphData["PentatopeGraph","Rigid"]

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

Flexible

GraphData["PentatopeGraph","Flexible"]

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

LCF

GraphData["PentatopeGraph","LCF"]

GraphData["PentatopeGraph","HamiltonianCycleCount"]

GraphData["PentatopeGraph","LCFSignature"]

GraphData["PentatopeGraph","LCFNotations"]

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

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

LCF compute all

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

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

lcf

LCFGraph@@@lcf//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["PentatopeGraph","UnitDistance"]

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

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

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

UnitDistance by refinement

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

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

UnitDistance by FindUnitDistanceEmbedding

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

Matchstick

GraphData["PentatopeGraph","Matchstick"]

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

3D default

Graph3D[GraphData["PentatopeGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["PentatopeGraph","Polyhedral"]

4D All

GraphData["PentatopeGraph","Graph",{"4D","All"}]

4D UnitDistance

GraphData["PentatopeGraph","Graph",{"4D","UnitDistance"}]

UnitDistanceGraphEmbeddingQ/@%

Construction

5-cell

v=1/(2 Sqrt[2]) {{2,0,0,0},{0,2,0,0},{0,0,2,0},{0,0,0,2},{GoldenRatio,GoldenRatio,GoldenRatio,GoldenRatio}};

ee=Subsets[Range[5],{2}];

g=Graph[Range@Length@v,UndirectedEdge@@@ee,VertexCoordinates->v]

r=RecognizeGraph@g

EdgeLengths[g]^2//FullSimplify

ToCommonEdges[g,GraphData[r]]

GraphData

GraphDataClassString

GraphDataClassString["PentatopeGraph"]