Trsat Graphs

Author

Eric W. Weisstein
April 30, 2025

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

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

<<MathWorld`Graphs`

"TrsatGraph1"

Names

Renamed from "U42Graph540" to "TrsatGraph1" in V14.3

GraphData["TrsatGraph1","StandardName"]

GraphData["TrsatGraph1","StandardNames"]

Primary

GraphData["TrsatGraph1"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["TrsatGraph1","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["TrsatGraph1","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["TrsatGraph1","EdgeList"]]

Default embedding

GraphPlot@GraphData["TrsatGraph1","Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData["TrsatGraph1","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["TrsatGraph1","Graph","Degenerate"]

Perspective

GraphData["TrsatGraph1","Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData["TrsatGraph1","Polyhedron"]

Bilateral

GraphData["TrsatGraph1","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["TrsatGraph1","Graph","Circular"]//StyleGraphs

Planar

GraphData["TrsatGraph1","Planar"]

GraphData["TrsatGraph1","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["TrsatGraph1",{"RCN","CN"}]

GraphData["TrsatGraph1","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["TrsatGraph1"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph1"],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["TrsatGraph1"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["TrsatGraph1"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["TrsatGraph1","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["TrsatGraph1"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["TrsatGraph1","Circulant"]

GraphData["TrsatGraph1","Graph","Circulant"]//StyleGraphs

Complement graph

GraphData["TrsatGraph1","ComplementGraph","Name"]

Line graph

GraphData["TrsatGraph1","Line"]

LineGraphQ[GraphData["TrsatGraph1"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "TrsatGraph1"&]

RootGraph[GraphData["TrsatGraph1"]]

Halin

GraphData["TrsatGraph1","Halin"]

GraphData["TrsatGraph1","Graph","Halin"]//StyleGraphs

Rigid

GraphData["TrsatGraph1","Rigid"]

GraphData["TrsatGraph1","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["TrsatGraph1","Flexible"]

GraphData["TrsatGraph1","Graph","Flexible"]//StyleGraphs

LCF

GraphData["TrsatGraph1","LCF"]

GraphData["TrsatGraph1","HamiltonianCycleCount"]

GraphData["TrsatGraph1","LCFSignature"]

GraphData["TrsatGraph1","LCFNotations"]

GraphData["TrsatGraph1","Graph","LCF"]//StyleGraphs

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

LCF compute random

FindRandomLCFNotation[GraphData["TrsatGraph1"],TimeConstraint->120]

FindRandomLCFNotation[GraphData["TrsatGraph1"],TimeConstraint->3600]

LCF compute 10^2

Tally[(lcf=LCF[GraphData["TrsatGraph1"],10^2])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

LCF compute 10^3

Tally[(lcf=LCF[GraphData["TrsatGraph1"],10^3])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

LCF compute 10^4

Tally[(lcf=LCF[GraphData["TrsatGraph1"],10^4])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

gs

LCF compute all

Divisors[GraphData["TrsatGraph1","V"]]

Length[FindHamiltonianCycle[GraphData["TrsatGraph1"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["TrsatGraph1"],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["TrsatGraph1",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

UnitDistance

GraphData["TrsatGraph1","UnitDistance"]

UnitDistanceGraphQ[GraphData["TrsatGraph1"],Debug->True]//Timing

GraphData["TrsatGraph1","Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["TrsatGraph1","VertexList"],GraphData["TrsatGraph1","EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["TrsatGraph1","Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["TrsatGraph1"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["TrsatGraph1"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData["TrsatGraph1","Matchstick"]

GraphData["TrsatGraph1","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["TrsatGraph1","Edges"]]

3D All

GraphData["TrsatGraph1","Graph",{"3D","All"}]

3D UnitDistance

GraphData["TrsatGraph1","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["TrsatGraph1","Polyhedral"]

GraphData["TrsatGraph1","Graph",{"3D","Polyhedron"}]

"TrsatGraph2"

Names

GraphData["TrsatGraph2","StandardName"]

GraphData["TrsatGraph2","StandardNames"]

Primary

GraphData["TrsatGraph2"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["TrsatGraph2","Graph","All"]//StyleGraphs

Construction from connected component

cc=GraphData["TrsatGraph2","ConnectedComponents","Name"]

GraphData[cc[[1]],"Graph","All"]

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

gs=MyGraphUnion[#,"K1"]&/@GraphData[cc[[1]],"Graph","All"]

RecognizeGraph/@gs

Inexact

Select[GraphData["TrsatGraph2","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embeddings

GraphDataEmbeddings[GraphData["TrsatGraph2"]]

Default embedding

GraphPlot[Graph[GraphData["TrsatGraph2","VertexList"],GraphData["TrsatGraph2","Edges"]]]

s=Sqrt[3]/2;
v={};
g=Graph[Range@Length@v,UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

g=FindCorrespondingGraph[,v]

RecognizeGraph@g

Degenerate

GraphData["TrsatGraph2","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["TrsatGraph2","Graph","Degenerate"]

Configuration

GetGraphData["TrsatGraph2","File"]

GraphData["TrsatGraph2","Configuration"]

GraphData["TrsatGraph2","Graph","Configuration"]//StyleGraphs

Configuration...

config=%[[1]]

(alledges=AddAllConfigurationEdges[config])//Timing

RecognizeGraph@alledges

ConfigurationDiagram[config,.2]

lines=ConfigurationLines[config]

ConfigurationNotation[lines]

GraphData["TrsatGraph2","IncidenceLines"]

levi=LeviGraph[lines]

RecognizeGraph[levi]

GraphData["TrsatGraph2","LeviGraph","Name"]

FirstCase[GraphData["TrsatGraph2","StandardNames"],{"Incidence",_}]

Cases[GraphData["TrsatGraph2","StandardNames"],_String?(StringMatchQ[#,"*Configuration*"]&)]

GraphData["TrsatGraph2","OrdinaryLineGraphName"]

(ordinarylinegraph=OrdinaryLineGraph[config])//Timing

RecognizeGraph@ordinarylinegraph

Incidence

GetGraphData["TrsatGraph2","File"]

GraphData["TrsatGraph2","Incidence"]

Select[GraphData[All],GraphData[#,"LeviGraph","Name"]==="TrsatGraph2"&]

lines=GraphData["TrsatGraph2","IncidenceLines"]

ConfigurationNotation[lines]

levi=LeviGraph[lines]

RecognizeGraph@levi

ProjectionGraph

Manipulate[ProjectionGraph[GraphData["TrsatGraph2","Graph","3D"],{a,b,c}],{{a,.5},0,1},{{b,.5},0,1},{{c,.5},0,1}]

gs=ProjectionGraph[GraphData["TrsatGraph2","Graph","3D"],#]&/@{{1/GoldenRatio,0,1},{0,0,1},{1,1,0},{1,1,1}}

Stereographic projection construction

gs=PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],#]&/@{3/2,4}//StyleGraphs

Perspective

GraphData["TrsatGraph2","Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData["TrsatGraph2","Polyhedron"]

PolyhedronGraph[GraphData["TrsatGraph2","Polyhedron"]]

Manipulate[PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{x,y,z}],{x,-1,1},{y,-1,1},{z,-1,1}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{0,.01,.01}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{0,0,1}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{0,1,0}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{1,0.01,0}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{1,1,0}]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{1,0,1},-Pi/2]

PolyhedronProjectionGraph[GraphData["TrsatGraph2","Polyhedron"],{1,1,1}]

HoneycombToroidal construction

GraphData["TrsatGraph2","NotationRules","HoneycombToroidal"]

HoneycombToroidal construction...

HoneycombToroidalGraph[1,,]

RecognizeGraph[%]

GeneralizedHoneycombGraph construction

With[{n=},Cases[Table[{k,g=GeneralizedHoneycombGraph[n,k],RecognizeGraph@g},{k,n-1}],{__,"TrsatGraph2"}]]//Timing

g=GeneralizedHoneycombGraph[,]

RecognizeGraph@g

GeneralizedPetersen

GraphData["TrsatGraph2","Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData["TrsatGraph2","Graph","IGraph"]//StyleGraphs

Bilateral

GraphData["TrsatGraph2","Graph","Bilateral"]//StyleGraphs

Grid[DeleteCases[Monitor[{g=#,BilateralGraphEmbeddingTypes[#],GraphSymmetryGroup[#]}&/@GraphData["TrsatGraph2","Graph","All"],g],{_,_,TransformationGroup[{}]}],Dividers->All,Alignment->{{Center,{Left}}}]

Circular

GraphData["TrsatGraph2","Graph","Circular"]//StyleGraphs

Circular construction

GraphData["TrsatGraph2","Hamiltonian"]

GraphData["TrsatGraph2","LCF"]

GraphData["TrsatGraph2","HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData["TrsatGraph2"],All][[All,All,1]]]//Timing

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData["TrsatGraph2"],#]&/@hc);//Timing

graphswithsymmetrygroups={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[graphswithsymmetrygroups[[All,-1]]]//Column

(bilaterallyrotatedgraphs={BilaterallyRotatedGraph[#,"ReturnNonbilateralGraph"->True],#2}&@@@SortBy[
DeleteCases[graphswithsymmetrygroups,{_,TransformationGroup[{}]}],
{MatchQ[#[[2]],TransformationGroup[{_Rotation,r_Reflection}|{r_Reflection|_Rotation}]],FirstCase[#[[2]],Rotation[{0,0},r_]:>N[r/π],{},{0,∞}]}&
])//Timing

Length[symmetricgraphs=DeleteDuplicatesBy[bilaterallyrotatedgraphs,CircularCanonicalGraphEmbedding[#[[1]]]&]]//Timing

symmetricgraphs

gs=symmetricgraphs[[All,1]]

Circular construction old

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph/@circ],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

Length[quad=DeleteDuplicatesBy[Union@Select[bilat,BilateralGraphEmbeddingQ[#,Axis->"X"]&],CircularCanonicalGraphEmbedding[#,4]&]]

quad

Planar

GraphData["TrsatGraph2","Planar"]

GraphData["TrsatGraph2","Graph","Planar"]//StyleGraphs

Planar construction

PlanarGraphEmbedding[GraphData["TrsatGraph2"],GraphLayout->#]&/@{"TutteLayout","DavidsonHarelLayout"}

Table[PlanarGraphEmbedding[GraphData["TrsatGraph2"]],10]

PlanarGraphEmbeddings

GraphData["TrsatGraph2","PlanarEmbeddingCount"]

(plemb=PlanarGraphEmbeddings[GraphData["TrsatGraph2"],Debug->True])//Timing

Length[plemb]

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["TrsatGraph2"]]

Tally[ReverseSort[Length/@faces]]

gs=IGLayoutTutte[GraphData["TrsatGraph2"],"OuterFace"->#]&/@faces

TextGrid[DeleteCases[{#,GraphSymmetryGroup[#]}&/@gs,{_,TransformationGroup[{}]}],Dividers->All]

Grid[DeleteCases[{#,Cases[GraphSymmetryGroup[#],r_Rotation:>r,∞]}&/@gs,{_,{}}],Dividers->All]

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

RecognizeGraph/@gs

Tutte embedding construction

GraphPlot[RotateGraph[]]

TutteLayoutManipulate[g_Graph]:=Manipulate[
With[{face=OuterFace[g]},
IGLayoutTutte[IGEdgeMap[(EuclideanDistance@@#)^pow&,EdgeWeight->IGEdgeVertexProp[VertexCoordinates],IGLayoutTutte[g,"OuterFace"->face]],"OuterFace"->face,ImageSize->Automatic]],{{pow,1},.1,2}]

TutteLayoutManipulate[]

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

RecognizeGraph[g]

MinimalCrossing

GraphData["TrsatGraph2",{"RCN","CN"}]

GraphData["TrsatGraph2","Graph","MinimalCrossing"]//StyleGraphs

MinimalRectilinearCrossing construction

Length[gs=FindMinimalRectilinearCrossingGraph[GraphData["TrsatGraph2"],Automatic,All,Method->"SubgraphIsomorphism",Debug->True]]//Timing

Tally[RectilinearCrossingCount/@gs]//Timing

GraphSymmetryGroup/@gs//Tally

Select[gs,GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]

gs

Length[gs=FindMinimalRectilinearCrossingGraph[GraphData["TrsatGraph2"],Automatic,All,Method->"Permutations",Debug->True]]//Timing

Tally[RectilinearCrossingCount/@gs]//Timing

GraphSymmetryGroup/@gs//Tally

Select[gs,GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]

gs

MinimalCrossingNumber: "EmbeddingScheme" -> 1

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->1,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 2

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->2,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossingNumber: "EmbeddingScheme" -> 3

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["TrsatGraph2"],"EmbeddingScheme"->3,TimeConstraint->86400])//AbsoluteTiming

MinimalCrossing construction

(cn=CrossingNumber[GraphData["TrsatGraph2"],TimeConstraint->1])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["TrsatGraph2"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Select[tuttegraphs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["TrsatGraph2","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Select[tuttegraphs,Count[OuterFace[#],_?(GreaterThan[GraphData["TrsatGraph2","V"]])]==0&]

Select[tuttegraphs,Count[OuterFace[#],_?(GreaterThan[GraphData["TrsatGraph2","V"]])]==1&]

(gs=Table[RotateGraph[MinimalCrossingGraph[GraphData["TrsatGraph2"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}]//Union)//Timing

Select[gs,Count[OuterFace[#],_?(GreaterThan[GraphData["TrsatGraph2","V"]])]==0&]

Select[gs,Count[OuterFace[#],_?(GreaterThan[GraphData["TrsatGraph2","V"]])]==1&]

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["TrsatGraph2","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

RectilinearLocalCrossingCount[g]

MinimalLocalCrossing

GraphData["TrsatGraph2",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["TrsatGraph2","Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalRectilinearLocalCrossing

GraphData["TrsatGraph2",{"LocalCrossingNumber","RectilinearLocalCrossingNumber"}]

GraphData["TrsatGraph2","Graph","MinimalRectilinearLocalCrossing"]//StyleGraphs

MinimaRectilinearlLocalCrossing construction

AdjacentCrossingsQ[g_Graph,cn_Integer]:=Module[{a=AdjacencyList[g],n=VertexCount[g]-cn},Max[Take[a,-cn]]>n]

(cn=CrossingNumber[GraphData["TrsatGraph2"],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData["TrsatGraph2"],"CrossingNumber"->cn,RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["TrsatGraph2","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

(gs=Select[Table[MinimalCrossingGraph[GraphData["TrsatGraph2"],"CrossingNumber"->(cn=XXX),RandomSeed->i],{i,20000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

Select[gs,GraphSymmetryGroup[#,AdditionalData->Join[Table[True,GraphData["TrsatGraph2","V"]],Table[False,cn]],SameTest->Equal]=!=TransformationGroup[{}]&]

Circulant

GraphData["TrsatGraph2","Circulant"]

GraphData["TrsatGraph2","Graph","Circulant"]//StyleGraphs

Circulant construction

GraphData["TrsatGraph2","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["TrsatGraph2"]])//Timing

CirculantGraph1@@@ci

Cases[GraphData["TrsatGraph2","StandardNames"],{"Circulant",i_}:>i]

CirculantGraph1@@@%

Complement graph

compl=GraphData["TrsatGraph2","ComplementGraph","Name"]

GraphData[compl,"Graph","All"]

gs=GraphComplement/@%

Select[%,!DegenerateGraphEmbeddingQ[#]&]

Line graph

GraphData["TrsatGraph2","Line"]

LineGraphQ[GraphData["TrsatGraph2"]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]==="TrsatGraph2"&]

RootGraph[GraphData["TrsatGraph2"]]

RecognizeGraph[%]

GraphData[roots[[1]],"Graph","All"]

gs=LineGraph/@%

Halin

GraphData["TrsatGraph2","Halin"]

GraphData["TrsatGraph2","Graph","Halin"]//StyleGraphs

Rigid

GraphData["TrsatGraph2","Rigid"]

GraphData["TrsatGraph2","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["TrsatGraph2","Flexible"]

GraphData["TrsatGraph2","Graph","Flexible"]//StyleGraphs

LCF

GraphData["TrsatGraph2","LCF"]

GraphData["TrsatGraph2","HamiltonianCycleCount"]

GraphData["TrsatGraph2","LCFSignature"]

GraphData["TrsatGraph2","LCFNotations"]

GraphData["TrsatGraph2","Graph","LCF"]//StyleGraphs

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

LCF compute random

FindRandomLCFNotation[GraphData["TrsatGraph2"],TimeConstraint->120]

LCF compute 10^5

Tally[(lcf=LCF[GraphData["TrsatGraph2"],10^5])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

(nice=SelectFirst[lcf,GraphSymmetryGroup[LCFGraph@@#]=!=TransformationGroup[{}]&])//Timing

l={
{"TrsatGraph2",{0,Missing["NotAvailable",LCFNotation/@gs]}}
}

PropertyReplace[l,"LCFNotations","14.3"]

RenameGraphFiles[]

LCF compute all

Divisors[GraphData["TrsatGraph2","V"]]

Length[FindHamiltonianCycle[GraphData["TrsatGraph2"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["TrsatGraph2"],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["TrsatGraph2",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2,1])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

(nice=SelectFirst[lcf,GraphSymmetryGroup[LCFGraph@@#]=!=TransformationGroup[{}]&])//Timing

i=0;
gs={};
t0=AbsoluteTime[];
count=Length[order2];
Monitor[
Do[
If[!MissingQ[g=BilateralLCFGraph[LCFGraph@@order2[[i]]]],AppendTo[gs,g]],
{i,count}
],
Column[{
ProgressIndicator[i,{1,count}],
Row[{i,"/",count}],
Row[{"estimated time remaining: ",Round[(AbsoluteTime[]-t0) (count/i-1)]," s"}],
Row[{Length[gs]," bilateral graphs"}]
}]
];//Timing

UnitDistance

GraphData["TrsatGraph2","UnitDistance"]

UnitDistanceGraphQ[GraphData["TrsatGraph2"],Debug->True]//Timing

GraphData["TrsatGraph2","Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["TrsatGraph2","VertexList"],GraphData["TrsatGraph2","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["TrsatGraph2","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["TrsatGraph2"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["TrsatGraph2"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance construction via UnitDistanceEmbeddingRefine of exact construction

(unitdistancegraphs=Flatten[With[{step=.1},
Table[
{{a,b,c},
UnitDistanceEmbeddingRefine@Graph[Range[40],ee,VertexCoordinates->N@Join[Sequence@@Table[CirclePoints[Through[{Cos,Sin}[Pi/2-2Pi i/5]],{a,Pi/2-2Pi i/5+2Pi i/6},6],{i,0,4}],CirclePoints[b,5],CirclePoints[c,5]]]},
{a,step,2,step},{b,step,2,step},{c,step,2,step}
]],2]);//Quiet//Timing

{Length[unitdistancegraphs],Count[unitdistancegraphs,_?(UnitDistanceGraphEmbeddingQ[Last[#]]&)]}

RandomSample[unitdistancegraphs,10]

Select[unitdistancegraphs,DegenerateGraphEmbeddingQ[Last[#]]===False&]//Timing

MinimumDistance[v_]:=Min[EuclideanDistance@@@Subsets[v,{2}]]

distances=MinimumDistance[GraphEmbedding[#]]&/@unitdistancegraphs[[All,-1]];

Histogram[distances]

Max[distances]

Position[distances,Max[distances]]

Take[ReverseSortBy[Transpose[{distances,unitdistancegraphs}],First],10]

UnitDistanceGraphEmbeddingQ/@gs

DegenerateGraphEmbeddingTypes/@gs

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"TrsatGraph2",l=FindSubgraphUnitDistanceEmbedding[GraphData["TrsatGraph2"],TimeConstraint->300,"Results"->All]//Timing}

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

Select[IndexGraph/@gs,!DegenerateGraphEmbeddingQ[#]&&GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]//Timing

Matchstick

GraphData["TrsatGraph2","Matchstick"]

GraphData["TrsatGraph2","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["TrsatGraph2","Edges"]]

3D construction

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

RecognizeGraph[g]

3D All

GraphData["TrsatGraph2","Graph",{"3D","All"}]

3D UnitDistance

GraphData["TrsatGraph2","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["TrsatGraph2","Polyhedral"]

GraphData["TrsatGraph2","Graph",{"3D","Polyhedron"}]

Construction

DistanceRegular.org

g1=DistanceRegularGraph["trsat1.am"]

RecognizeGraph@g1

g2=DistanceRegularGraph["trsat2.am"]

IsomorphicGraphQ[g2,g1]

IsomorphicGraphQ[g2,GraphComplement@g1]

Sage

from sage.features.gap import GapPackage
GapPackage("grape", spkg="gap_packages").is_present()
G=graphs.U42Graph540()
G.edges()

data=;

g=CanonicalGraph[Graph[UndirectedEdge@@@Partition[StringCases[data,DigitCharacter..],2]]]

RecognizeGraph@g

RegularParameters[g]

GraphData

GraphDataClassString

GraphDataClassString[{"TrsatGraph1","TrsatGraph2"}]

GraphDataString

"TrsatGraph1"

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

WriteString[file,GraphDataString[g,
"StandardName"->"U42Graph540",
"Name"->"U_4(2) 540\[Hyphen]graph",
"Information"->None,
"Embeddings"->{},
"Skip"->{"Arboricity","BurningNumber","Chord","ChordlessCycle","ChordlessCycles","Chords","ChromaticNumber","CliqueCoveringNumber","ComplementOddChordlessCycle","ComplementOddChordlessCycles","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","CrossingNumber","Cycles","CyclePolynomial","DominatingSet","DominatingSets","EdgeCovers","EdgeCuts","GracefulLabeling","GracefulLabelings","HamiltonianCycles","HamiltonianPaths","HamiltonianWalks","IdiosyncraticPolynomial","IndependentVertexSets","LongestCycles","MatchingGeneratingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSets","MaximallyNonhamiltonian","MaximumFlowMatrix","MaximumIndependentEdgeSets","MinimumCoveringsByMaximalCliques","MinimumEdgeColoring","MinimumVertexColoring","MinimumVertexCover","MinimumVertexCut","MinimumVertexCuts","MinimumWeightFractionalColoring","MinimumWeightFractionalEdgeColoring","OddChordlessCycles","PathPolynomial","Paths","RadioLabeling","RadioLabelings","RadioNumber","RectilinearCrossingCount","ResistanceMatrix","Skewness","StronglyPerfect","UnitDistance","VertexCover","Thickness"},
TimeConstraint->10
]]

Close[file]

"TrsatGraph2"

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

WriteString[
file,
GraphDataString[
,
"Embeddings"->{},
"StandardName"->"TrsatGraph2",
"Name"->"second Trsat graph",
"Information"->"TrsatGraphs",
"Skip"->{"ABCEnergy","ABCIndex","ABCSpectralRadius","Anarboricity","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","ChordlessCycles","Chords","ChromaticNumber","CirculantIndices","CliqueCoveringNumber","ComplementChordlessCycles","ConnectedInducedSubgraphPolynomial","ConstructionNumber","CyclicEdgeConnectivity","CyclicEdgeCuts","DetourPolynomial","DistinguishingNumber","DominatingSets","EdgeBetweennessCentralities","EdgeChromaticNumber","EdgeCovers","EdgeCuts","Energy","ErdosSequence","FractionalIndependenceNumber","GracefulLabeling","GracefulLabelings","HadwigerNumber","IdiosyncraticPolynomial","IndependenceNumber","IndependencePolynomial","IndependentEdgeSets","IndependentVertexSets","IntersectionNumber","IrredundancePolynomial","LambdaComponents","LaplacianPolynomial","MatchingGeneratingPolynomial","MaximalIndependencePolynomial","MaximalIndependentEdgeSet","MaximumIndependentEdgeSets","MaximumIndependentEdgeSetCount","MaximalIndependentVertexSet","MaximumFlowMatrix","MaximumIndependentVertexSet","MaximumIndependentVertexSetCount","MaximumIndependentVertexSets","MaximalIndependentVertexSets","MetricDimension","Meyniel","MinimumConnectedDominatingSet","MinimumConnectedDominatingSets","MinimumCostFlowMatrix","MinimumCoveringsByMaximalCliques","MinimumCyclicEdgeCuts","MinimumDominatingSets","MinimumEdgeColoring","MaximumFlowMatrix","MinimumTotalDominatingSets","MinimumVertexColoring","MinimumVertexColorings","MinimumVertexCover","MinimumVertexCut","MinimumWeightFractionalColoring","MinimumWeightFractionalEdgeColoring","Minors","QuadraticallyEmbeddable","QuadraticEmbeddingConstant","RandicEnergy","ResistanceMatrix","SomborEnergy","SomborSpectralRadius","TotalDominationPolynomial","TotalDominatingSets","TuttePolynomial","VertexConnectivity"},
TimeConstraint->10
]]

Close[file]