Dejter Graph

Author

Eric W. Weisstein
November 10, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Sources

<<MathWorld`Graphs`

Names

GraphData["DejterGraph","StandardName"]

GraphData["DejterGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["DejterGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Perspective

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

Perspective construction

GraphData["DejterGraph","Polyhedron"]

Bilateral

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

Circular

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

Planar

GraphData["DejterGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["DejterGraph","Circulant"]

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

Complement graph

GraphData["DejterGraph","ComplementGraph","Name"]

GraphComplement/@GraphData[%,"Graph","All"]

Line graph

GraphData["DejterGraph","Line"]

LineGraphQ[GraphData["DejterGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["DejterGraph"]]

Halin

GraphData["DejterGraph","Halin"]

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

Rigid

GraphData["DejterGraph","Rigid"]

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

Flexible

GraphData["DejterGraph","Flexible"]

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

LCF

GraphData["DejterGraph","LCF"]

GraphData["DejterGraph","HamiltonianCycleCount"]

GraphData["DejterGraph","LCFSignature"]

GraphData["DejterGraph","LCFNotations"]

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

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

LCF compute random

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

LCF compute 10^4

Tally[(lcf=LCF[GraphData["DejterGraph"],10^4])[[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 10^5

Tally[(lcf=LCF[GraphData["DejterGraph"],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]

LCF compute 10^6

Tally[(lcf=LCF[GraphData["DejterGraph"],10^6])[[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 10^7

Tally[(lcf=LCF[GraphData["DejterGraph"],10^7])[[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 all

Divisors[GraphData["DejterGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["DejterGraph",".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["DejterGraph","UnitDistance"]

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

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

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

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

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

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

Matchstick

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

3D default

Graph3D[GraphData["DejterGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["DejterGraph","Polyhedral"]

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

Construction

Wikipedia

https://en.wikipedia.org/wiki/Dejter_graph

Sage

https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html#sage.graphs.graph_generators.GraphGenerators.DejterGraph

g=SageGraph[SageEvaluate["graphs.DejterGraph().edges()"]]

Table[CycleCount[g,n],{n,6}]

Through[{VertexCount,EdgeCount,GraphRadius,GraphDiameter,Girth,ChromaticNumber}[g]]

Wikipedia claims girth = 6 ??

GroupOrder[GraphAutomorphismGroup[g]]

RegularParameters[g]//Timing

<<IGraphM`

IGDistanceRegularQ[g]

GraphData["Q7","V"]

BipartiteGraphQ[g]

coloring=FindVertexColoring[g,Debug->True]

biparts=SplitBy[Sort[MapIndexed[{#,#2[[1]]}&,coloring]],First][[All,All,2]]

HighlightGraph[g,{Style[biparts[[1]],Red],Style[biparts[[2]],Blue]},VertexLabels->"Name"]

GraphData

GraphDataClassString

GraphDataClassString[g]

GraphDataString

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

WriteString[file,GraphDataString[g,
"StandardName"->"DejterGraph",
"Name"->"Dejter graph",
"Information"->"DejterGraph",
"Embeddings"->{},
"Skip"->{"Arboricity","BurningNumber","Chord","ChordlessCycle","ChordlessCycles","Chords","ChromaticNumber","CliqueCoveringNumber","ComplementOddChordlessCycle","ComplementOddChordlessCycles","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","CrossingNumber","Cycles","CyclePolynomial","DominatingSet","DominatingSets","EdgeCovers","GracefulLabeling","GracefulLabelings","HamiltonianCycles","HamiltonianPaths","HamiltonianWalks","IdiosyncraticPolynomial","IndependentVertexSets","LongestCycles","MatchingGeneratingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSets","MaximallyNonhamiltonian","MaximumFlowMatrix","MaximumIndependentEdgeSets","MinimumCoveringsByMaximalCliques","MinimumEdgeColoring","MinimumVertexColoring","MinimumVertexCut","MinimumVertexCuts","MinimumWeightFractionalColoring","MinimumWeightFractionalEdgeColoring","OddChordlessCycles","PathPolynomial","Paths","RadioLabeling","RadioLabelings","RadioNumber","RectilinearCrossingCount","ResistanceMatrix","Skewness","StronglyPerfect","UnitDistance","Thickness"},
TimeConstraint->10
]]

Close[file]