Dorogovtsev-Goltsev-Mendes Graph

Author

Eric W. Weisstein
January 20, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Dorogovtsev-Goltsev-MendesGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Dorogovtsev-Goltsev-MendesGraph.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Source

S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes.  "Pseudofractal Scale-free Web."  8 Dec 2001.  https://arxiv.org/abs/cond-mat/0112143

Classes

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["DorogovtsevGoltsevMendes"])

GraphClassesThatAreClass["DorogovtsevGoltsevMendes"]

Figure

gs=ReplacePart[
Table[GraphData[{"DorogovtsevGoltsevMendes",n}],{n,0,4}],
{3->GraphData[{"TriangularGrid", 2},"Graph",2]}]

GraphicsRow[Table[Show[StyleGraphs@gs[[n+1]],PlotLabel->Style[Row[{Style["n",Italic]," == ",n}],Directive[14,Black,FontFamily->"Times"]]],{n,0,4}],Dividers->All,Alignment->Top,ImageSize->800]

Embeddings

<<MathWorld`Graphs`

{"DorogovtsevGoltsevMendes", 0} == {"Path", 2}

Names

GraphData[{"DorogovtsevGoltsevMendes",0},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",0},"StandardNames"]

{"DorogovtsevGoltsevMendes", 1} == "TriangleGraph"

Names

GraphData[{"DorogovtsevGoltsevMendes",1},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",1},"StandardNames"]

{"DorogovtsevGoltsevMendes", 2} == {"TriangularGrid", 2}

Names

GraphData[{"DorogovtsevGoltsevMendes",2},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",2},"StandardNames"]

{"DorogovtsevGoltsevMendes", 3}

Names

GraphData[{"DorogovtsevGoltsevMendes",3},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",3},"StandardNames"]

Primary

GraphData[{"DorogovtsevGoltsevMendes",3}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"DorogovtsevGoltsevMendes",3},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"DorogovtsevGoltsevMendes",3},"Edges"]

v=UnitCirclePoints[3,-Pi/2];
v=Join[v,Sequence@@Table[RotationMatrix[2Pi i/3].#&/@(v[[#]]+{0,1}&/@{2,3}),{i,0,2}]];
v=Join[v,Mean[v[[#]]]&/@{{4,5},{8,9},{6,7}}];
v=Join[v,Mean[v[[#]]]&/@{{2,3,10},{1,11,2},{1,3,12}}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,8,11,9,2,4,10,5,3,6,12,7,1,14,2,13,3,15,1,2,3,1,11,2,10,3,12,1]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Construction from Dorogovtsev et al.

v=UnitCirclePoints[6,-Pi/2];
v=Join[v,v[[#]]-{0,1}&/@{2,6},v[[#]]+(v[[2]]-v[[1]])&/@{2,4},v[[#]]+(v[[4]]-v[[3]])&/@{6,4}];
v=Join[v,v[[#]]+{0,1}&/@{11,9},{v[[8]]+v[[1]]-v[[6]]}];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[1,2,3,4,5,6,1],e[15,7,2,9,14,10,4,12,13,11,6,8,15,2,14,4,13,6,15],e[2,4,6,2]},VertexCoordinates->v,VertexLabels->Automatic]

IsomorphicGraphQ[g,DorogovtsevGoltsevMendesGraph[3]]

Degenerate

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Degenerate"]

Perspective

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"DorogovtsevGoltsevMendes",3},"Polyhedron"]

Bilateral

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"DorogovtsevGoltsevMendes",3},"Planar"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"DorogovtsevGoltsevMendes",3},{"RCN","CN"}]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",3}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",3}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",3}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",3}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"DorogovtsevGoltsevMendes",3},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",3}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

GraphPlot@

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"DorogovtsevGoltsevMendes",3},"Circulant"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"DorogovtsevGoltsevMendes",3},"ComplementGraph","Name"]

Line graph

GraphData[{"DorogovtsevGoltsevMendes",3},"Line"]

LineGraphQ[GraphData[{"DorogovtsevGoltsevMendes",3}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"DorogovtsevGoltsevMendes",3}&]

RootGraph[GraphData[{"DorogovtsevGoltsevMendes",3}]]

Halin

GraphData[{"DorogovtsevGoltsevMendes",3},"Halin"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"DorogovtsevGoltsevMendes",3},"Rigid"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"DorogovtsevGoltsevMendes",3},"Flexible"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"DorogovtsevGoltsevMendes",3},"LCF"]

GraphData[{"DorogovtsevGoltsevMendes",3},"HamiltonianCycleCount"]

GraphData[{"DorogovtsevGoltsevMendes",3},"LCFSignature"]

GraphData[{"DorogovtsevGoltsevMendes",3},"LCFNotations"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",3},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"DorogovtsevGoltsevMendes",3}],Debug->True]//Timing

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"DorogovtsevGoltsevMendes",3},"VertexList"],GraphData[{"DorogovtsevGoltsevMendes",3},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"DorogovtsevGoltsevMendes",3},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",3}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",3}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"DorogovtsevGoltsevMendes",3},"Matchstick"]

GraphData[{"DorogovtsevGoltsevMendes",3},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"DorogovtsevGoltsevMendes",3},"Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData[{"DorogovtsevGoltsevMendes",3},"Polyhedral"]

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

{"DorogovtsevGoltsevMendes", 4}

Names

GraphData[{"DorogovtsevGoltsevMendes",4},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",4},"StandardNames"]

Primary

GraphData[{"DorogovtsevGoltsevMendes",4}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"DorogovtsevGoltsevMendes",4},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"DorogovtsevGoltsevMendes",4},"Edges"]

LabelGraph@

ExtendPerp[v[[{1,3}]],.5]

EuclideanDistance@@v[[{1,3}]]

ExtendPerp[vs:{v1_,v2_},a_]:=Module[{m=Mean[vs],perp={-#2,#}&@@Normalize[Subtract@@vs]},
m+a perp
]

s=1/Sqrt[3];
v={{1/2,s/2},{0,s},{0,3s/2},{0,3s/4}};
v=Join[v,{ExtendPerp[v[[{1,2}]],1/16],UnitDistancePoint[v[[1]],v[[3]]]}];
v=Join[v,ExtendPerp[v[[#]],1/4]&/@{{6,3},{1,6}}];
v=Join[v,{Mean[v[[{1,3,6}]]]}];
v1=DeleteDuplicates@RootReduce@Join[v,{-#,#2}&@@@v];
v2=DeleteDuplicates@RootReduce@Flatten[Table[RotationMatrix[2Pi i/3].#&/@v1,{i,0,2}],1];
g=Graph[Range[Length[v2]],UndirectedEdge@@@{e[24,36,34,35,31,40,39,41,1,8,6,7,3,13,12,14,10,22,20,21,17,27,26,28,24,1,10,24,32,1,4,10,18,24,30,1,2,10,16,24,33,30,38,1,5,2,11,10,19,16,25,24,37,31,42,1,9,3,15,10,23,17,29,24,34,31,39,1,6,3,12,10,20,17,26,24,31,1,3,10,17,24]},VertexCoordinates->v2,VertexLabels->Automatic,Axes->True]

RecognizeGraph@g

PlanarGraphEmbeddingQ@g

Degenerate

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Degenerate"]

Perspective

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"DorogovtsevGoltsevMendes",4},"Polyhedron"]

Bilateral

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"DorogovtsevGoltsevMendes",4},"Planar"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"DorogovtsevGoltsevMendes",4},{"RCN","CN"}]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",4}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",4}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",4}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",4}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"DorogovtsevGoltsevMendes",4},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",4}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"DorogovtsevGoltsevMendes",4},"Circulant"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"DorogovtsevGoltsevMendes",4},"ComplementGraph","Name"]

Line graph

GraphData[{"DorogovtsevGoltsevMendes",4},"Line"]

LineGraphQ[GraphData[{"DorogovtsevGoltsevMendes",4}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"DorogovtsevGoltsevMendes",4}&]

RootGraph[GraphData[{"DorogovtsevGoltsevMendes",4}]]

Halin

GraphData[{"DorogovtsevGoltsevMendes",4},"Halin"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"DorogovtsevGoltsevMendes",4},"Rigid"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"DorogovtsevGoltsevMendes",4},"Flexible"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"DorogovtsevGoltsevMendes",4},"LCF"]

GraphData[{"DorogovtsevGoltsevMendes",4},"HamiltonianCycleCount"]

GraphData[{"DorogovtsevGoltsevMendes",4},"LCFSignature"]

GraphData[{"DorogovtsevGoltsevMendes",4},"LCFNotations"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",4},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"DorogovtsevGoltsevMendes",4}],Debug->True]//Timing

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"DorogovtsevGoltsevMendes",4},"VertexList"],GraphData[{"DorogovtsevGoltsevMendes",4},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"DorogovtsevGoltsevMendes",4},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",4}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",4}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"DorogovtsevGoltsevMendes",4},"Matchstick"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"DorogovtsevGoltsevMendes",4},"Edges"]]

3D All

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"DorogovtsevGoltsevMendes",4},"Polyhedral"]

GraphData[{"DorogovtsevGoltsevMendes",4},"Graph",{"3D","Polyhedron"}]

{"DorogovtsevGoltsevMendes", 5}

Names

GraphData[{"DorogovtsevGoltsevMendes",5},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",5},"StandardNames"]

Primary

GraphData[{"DorogovtsevGoltsevMendes",5}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"DorogovtsevGoltsevMendes",5},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"DorogovtsevGoltsevMendes",5},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Degenerate"]

Perspective

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"DorogovtsevGoltsevMendes",5},"Polyhedron"]

Bilateral

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"DorogovtsevGoltsevMendes",5},"Planar"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"DorogovtsevGoltsevMendes",5},{"RCN","CN"}]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",5}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",5}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",5}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",5}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"DorogovtsevGoltsevMendes",5},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",5}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"DorogovtsevGoltsevMendes",5},"Circulant"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"DorogovtsevGoltsevMendes",5},"ComplementGraph","Name"]

Line graph

GraphData[{"DorogovtsevGoltsevMendes",5},"Line"]

LineGraphQ[GraphData[{"DorogovtsevGoltsevMendes",5}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"DorogovtsevGoltsevMendes",5}&]

RootGraph[GraphData[{"DorogovtsevGoltsevMendes",5}]]

Halin

GraphData[{"DorogovtsevGoltsevMendes",5},"Halin"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"DorogovtsevGoltsevMendes",5},"Rigid"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"DorogovtsevGoltsevMendes",5},"Flexible"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"DorogovtsevGoltsevMendes",5},"LCF"]

GraphData[{"DorogovtsevGoltsevMendes",5},"HamiltonianCycleCount"]

GraphData[{"DorogovtsevGoltsevMendes",5},"LCFSignature"]

GraphData[{"DorogovtsevGoltsevMendes",5},"LCFNotations"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",5},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"DorogovtsevGoltsevMendes",5}],Debug->True]//Timing

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"DorogovtsevGoltsevMendes",5},"VertexList"],GraphData[{"DorogovtsevGoltsevMendes",5},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"DorogovtsevGoltsevMendes",5},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",5}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",5}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"DorogovtsevGoltsevMendes",5},"Matchstick"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"DorogovtsevGoltsevMendes",5},"Edges"]]

3D All

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"DorogovtsevGoltsevMendes",5},"Polyhedral"]

GraphData[{"DorogovtsevGoltsevMendes",5},"Graph",{"3D","Polyhedron"}]

{"DorogovtsevGoltsevMendes", 6}

Names

GraphData[{"DorogovtsevGoltsevMendes",6},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",6},"StandardNames"]

Primary

GraphData[{"DorogovtsevGoltsevMendes",6}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"DorogovtsevGoltsevMendes",6},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"DorogovtsevGoltsevMendes",6},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Degenerate"]

Perspective

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"DorogovtsevGoltsevMendes",6},"Polyhedron"]

Bilateral

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"DorogovtsevGoltsevMendes",6},"Planar"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"DorogovtsevGoltsevMendes",6},{"RCN","CN"}]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",6}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",6}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",6}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",6}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"DorogovtsevGoltsevMendes",6},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",6}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"DorogovtsevGoltsevMendes",6},"Circulant"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"DorogovtsevGoltsevMendes",6},"ComplementGraph","Name"]

Line graph

GraphData[{"DorogovtsevGoltsevMendes",6},"Line"]

LineGraphQ[GraphData[{"DorogovtsevGoltsevMendes",6}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"DorogovtsevGoltsevMendes",6}&]

RootGraph[GraphData[{"DorogovtsevGoltsevMendes",6}]]

Halin

GraphData[{"DorogovtsevGoltsevMendes",6},"Halin"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"DorogovtsevGoltsevMendes",6},"Rigid"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"DorogovtsevGoltsevMendes",6},"Flexible"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"DorogovtsevGoltsevMendes",6},"LCF"]

GraphData[{"DorogovtsevGoltsevMendes",6},"HamiltonianCycleCount"]

GraphData[{"DorogovtsevGoltsevMendes",6},"LCFSignature"]

GraphData[{"DorogovtsevGoltsevMendes",6},"LCFNotations"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",6},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"DorogovtsevGoltsevMendes",6}],Debug->True]//Timing

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"DorogovtsevGoltsevMendes",6},"VertexList"],GraphData[{"DorogovtsevGoltsevMendes",6},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"DorogovtsevGoltsevMendes",6},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",6}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",6}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"DorogovtsevGoltsevMendes",6},"Matchstick"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"DorogovtsevGoltsevMendes",6},"Edges"]]

3D All

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"DorogovtsevGoltsevMendes",6},"Polyhedral"]

GraphData[{"DorogovtsevGoltsevMendes",6},"Graph",{"3D","Polyhedron"}]

{"DorogovtsevGoltsevMendes", 7}

Names

GraphData[{"DorogovtsevGoltsevMendes",7},"StandardName"]

GraphData[{"DorogovtsevGoltsevMendes",7},"StandardNames"]

Primary

GraphData[{"DorogovtsevGoltsevMendes",7}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"DorogovtsevGoltsevMendes",7},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"DorogovtsevGoltsevMendes",7},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Degenerate"]

Perspective

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"DorogovtsevGoltsevMendes",7},"Polyhedron"]

Bilateral

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"DorogovtsevGoltsevMendes",7},"Planar"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"DorogovtsevGoltsevMendes",7},{"RCN","CN"}]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",7}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"DorogovtsevGoltsevMendes",7}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",7}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",7}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"DorogovtsevGoltsevMendes",7},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"DorogovtsevGoltsevMendes",7}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"DorogovtsevGoltsevMendes",7},"Circulant"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"DorogovtsevGoltsevMendes",7},"ComplementGraph","Name"]

Line graph

GraphData[{"DorogovtsevGoltsevMendes",7},"Line"]

LineGraphQ[GraphData[{"DorogovtsevGoltsevMendes",7}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"DorogovtsevGoltsevMendes",7}&]

RootGraph[GraphData[{"DorogovtsevGoltsevMendes",7}]]

Halin

GraphData[{"DorogovtsevGoltsevMendes",7},"Halin"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"DorogovtsevGoltsevMendes",7},"Rigid"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"DorogovtsevGoltsevMendes",7},"Flexible"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"DorogovtsevGoltsevMendes",7},"LCF"]

GraphData[{"DorogovtsevGoltsevMendes",7},"HamiltonianCycleCount"]

GraphData[{"DorogovtsevGoltsevMendes",7},"LCFSignature"]

GraphData[{"DorogovtsevGoltsevMendes",7},"LCFNotations"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",7},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"DorogovtsevGoltsevMendes",7}],Debug->True]//Timing

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","UnitDistance"]//StyleGraphs

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"DorogovtsevGoltsevMendes",7},"VertexList"],GraphData[{"DorogovtsevGoltsevMendes",7},"EdgeList"]],MaxIterations->500]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"DorogovtsevGoltsevMendes",7},"Graph",#]&/@{"All","UnitDistance"})

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",7}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"DorogovtsevGoltsevMendes",7}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData[{"DorogovtsevGoltsevMendes",7},"Matchstick"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"DorogovtsevGoltsevMendes",7},"Edges"]]

3D All

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"DorogovtsevGoltsevMendes",7},"Polyhedral"]

GraphData[{"DorogovtsevGoltsevMendes",7},"Graph",{"3D","Polyhedron"}]

Properties

Acyclic

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Acyclic"],{n,0,7}]

AdjacencyMatrixCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"AdjacencyMatrixCount"],{n,0,7}]

AlgebraicConnectivity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"AlgebraicConnectivity"],{n,0,7}]

AlmostHamiltonian

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"AlmostHamiltonian"],{n,0,7}]

Anarboricity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Anarboricity"],{n,0,7}]

Antipodal

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Antipodal"],{n,0,7}]

Apex

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Apex"],{n,0,7}]

ApexCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ApexCount"],{n,0,7}]

gf=FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n≥0]

Arboricity

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Arboricity"],{n,0,7}]

ArcTransitive

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ArcTransitive"],{n,0,7}]

ArcTransitivity

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ArcTransitivity"],{n,0,7}]

ArticulationVertexCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ArticulationVertexCount"],{n,0,7}]

AutomorphismCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"AutomorphismCount"],{n,0,7}]

BalabanIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BalabanIndex"],{n,0,7}]

Bandwidth

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Bandwidth"],{n,0,7}]

Bicolorable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Bicolorable"],{n,0,7}]

Biconnected

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Biconnected"],{n,0,7}]

Bicubic

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Bicubic"],{n,0,7}]

Bipartite

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Bipartite"],{n,0,7}]

BipartiteDimension

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BipartiteDimension"],{n,0,7}]

Biplanar

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Biplanar"],{n,0,7}]

Block

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Block"],{n,0,7}]

BlockCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BlockCount"],{n,0,7}]

BridgeCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BridgeCount"],{n,0,7}]

Bridged

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Bridged"],{n,0,7}]

BroadcastTime

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BroadcastTime"],{n,0,7}]

BurningNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"BurningNumber"],{n,0,7}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CharacteristicPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

CheegerConstant

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CheegerConstant"],{n,0,7}]

Chordal

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Chordal"],{n,0,7}]

ChordCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChordCount"],{n,0,7}]

ChordlessCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChordlessCycleCount"],{n,0,7}]

ChordlessCyclePolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChordlessCyclePolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

ChromaticInvariant

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChromaticInvariant"],{n,0,7}]

FindSequenceFunction[seq,n+1]

ChromaticNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChromaticNumber"],{n,0,7}]

ChromaticPolynomial

Inferred for n > 4:

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ChromaticPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

Table[(((3^n)-1)/2),{n,0,7}]

CircuitRank

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CircuitRank"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

Circumference

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Circumference"],{n,0,7}]

FindSequenceFunction[seq,n+1]

Classes

Intersection@@(GraphData[#,"Classes"]&/@GraphData["DorogovtsevGoltsevMendes"])

ClawFree

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ClawFree"],{n,0,7}]

CliqueCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CliqueCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

CliqueCoveringNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CliqueCoveringNumber"],{n,0,7}]

SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}]

CliqueNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CliqueNumber"],{n,0,7}]

CliquePolynomial

(poly=Factor/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CliquePolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Coarseness

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Coarseness"],{n,0,7}]

CoboundaryPolynomial

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CoboundaryPolynomial"],{n,0,7}]

ComplementChordlessCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ComplementChordlessCycleCount"],{n,0,7}]

ComplementChordlessCyclePolynomial

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ComplementChordlessCyclePolynomial"],{n,0,7}]/.f_Function:>f[x]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ComplementOddChordlessCycleCount"],{n,0,7}]

ComplementOddChordlessCyclePolynomial

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ComplementOddChordlessCyclePolynomial"],{n,0,7}]/.f_Function:>f[x]

Conductance

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Conductance"],{n,0,7}]

Connected

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Connected"],{n,0,7}]

ConnectedComponentCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedComponentCount"],{n,0,7}]

ConnectedDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedDominatingSetCount"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedDominationNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedDominationNumber"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedDominationPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedDominationPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedInducedSubgraphCount"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n]

ConnectedInducedSubgraphPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Corank

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Corank"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n+1]//FullSimplify

CriticalNonplanar

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CriticalNonplanar"],{n,0,7}]

CrossingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CrossingNumber"],{n,0,7}]

Cubic

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CriticalNonplanar"],{n,0,7}]

CycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CycleCount"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n+1]

CyclePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CyclePolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

Cyclic

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Cyclic"],{n,0,7}]

CyclicEdgeConnectivity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"CyclicEdgeConnectivity"],{n,0,7}]

Degeneracy

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Degeneracy"],{n,0,7}]

DegreeCentralities

Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DegreeCentralities"],{n,0,7}]//Column

DetourIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DetourIndex"],{n,0,7}]

FindSequenceFunction[seq,n+1]//Factor

DetourPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DetourPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Diameter

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Diameter"],{n,0,7}]

Dimension

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Dimension"],{n,0,7}]

DistanceHereditary

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DistanceHereditary"],{n,0,7}]

DistancePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DistancePolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

DistanceRegular

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DistanceRegular"],{n,0,7}]

DistanceTransitive

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DistanceTransitive"],{n,0,7}]

DistinguishingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DistinguishingNumber"],{n,0,7}]

DomaticNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DomaticNumber"],{n,0,7}]

DominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DominatingSetCount"],{n,0,7}]

DominationNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DominationNumber"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>1]

DominationPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"DominationPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Doublecross

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Doublecross"],{n,0,7}]

Eccentricities

(Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Eccentricities"],{n,0,7}])//Column

EccentricityCentralities

(Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EccentricityCentralities"],{n,0,7}])//Column

EdgeBetweennessCentralities

(Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeBetweennessCentralities"],{n,0,7}])//Column

EdgeChromaticNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeChromaticNumber"],{n,0,7}]

EdgeConnectivity

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeConnectivity"],{n,0,7}]

EdgeCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

EdgeCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeCoverCount"],{n,0,7}]

EdgeCoverNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeCoverNumber"],{n,0,7}]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeCoverPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

EdgeCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeCoverNumber"],{n,0,7}]

EdgeTransitive

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EdgeTransitive"],{n,0,7}]

EigenvectorCentralities

(Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EigenvectorCentralities"],{n,0,7}])//Column

EulerianCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"EulerianCycleCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

FaceCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FaceCount"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n≥0]

FixingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FixingNumber"],{n,0,7}]

FlowPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FlowPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

FractionalArboricity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FractionalArboricity"],{n,0,7}]

FractionalChromaticNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FractionalChromaticNumber"],{n,0,7}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FractionalEdgeChromaticNumber"],{n,0,7}]

FractionaIndependenceNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"FractionalIndependenceNumber"],{n,0,7}]

Genus

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Genus"],{n,0,7}]

Geodetic

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Geodetic"],{n,0,7}]

Girth

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Girth"],{n,0,7}]

GlobalClusteringCoefficient

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"GlobalClusteringCoefficient"],{n,0,7}]

FindSequenceFunction[Rest@seq,n]

Gonality

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Gonality"],{n,0,7}]

Graceful

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Graceful"],{n,0,7}]

GracefulLabelingCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"GracefulLabelingCount"],{n,0,7}]

HadwigerNumber

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HadwigerNumber"],{n,0,7}]

HaemersNumber

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HaemersNumber"],{n,0,7}]

HamiltonConnected

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonConnected"],{n,0,7}]

HamiltonDecomposable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonDecomposable"],{n,0,7}]

HamiltonDecompositionCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonDecompositionCount"],{n,0,7}]

Hamiltonian

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Hamiltonian"],{n,0,7}]

HamiltonianCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonianCycleCount"],{n,0,7}]

HamiltonianNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonianNumber"],{n,0,7}]

HamiltonianPathCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonianPathCount"],{n,0,7}]

FindSequenceFunction[Drop[seq,1],n-1]

HamiltonianWalkCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonianWalkCount"],{n,0,7}]

HamiltonLaceable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HamiltonianWalkCount"],{n,0,7}]

HararyIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HararyIndex"],{n,0,7}]

HexagonCount

A367968

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HexagonCount"],{n,0,7}]

FindGeneratingFunction[seq,x]//Head//Timing

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CycleCount[DorogovtsevGoltsevMendesGraph[n],6]//Timing,MaxMemoryUsed[]/1.*^9}],{n,10}];

seq={0,0,1,38,276,1270,4745,15936,50608,156116,474585};

gf=FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n≥0]//Factor

Table[(65 (3^n)-84 n-6 (n^2)-40 (n^3)-65)/8,{n,0,20}]

FindLinearRecurrence[seq]

seq=LinearRecurrence[{7,-18,22,-13,3},{0,0,1,38,276},27]

ToString[seq,InputForm]//StringLength

Cf. A003462(n) (3-cycles), A290764(n-1) (4-cycles), A367967 (5-cycles), A367968 (6-cycles).

HosoyaIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HosoyaIndex"],{n,0,7}]

HStarConnected

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"HStarConnected"],{n,0,7}]

Hypohamiltonian

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Hypohamiltonian"],{n,0,7}]

Hypotraceable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Hypotraceable"],{n,0,7}]

Identity

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Identity"],{n,0,7}]

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IdiosyncraticPolynomial"],{n,k0=0,7}]/.f_Function:>f[x,y])//Column

IndependenceNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IndependenceNumber"],{n,0,7}]

IndependencePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IndependencePolynomial"],{n,k0=0,7}]/.f_Funtion:>f[x])/.f_Function:>f[x]//Column

IndependenceRatio

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IndependenceRatio"],{n,0,7}]

FindSequenceFunction[seq,n+1]

IndependentEdgeSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IndependentEdgeSetCount"],{n,0,7}]

IndependentVertexSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IndependentVertexSetCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

IntersectionNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IntersectionNumber"],{n,0,7}]

IrredundanceNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IrredundanceNumber"],{n,0,7}]

IrredundancePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IrredundancePolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

IrredundantSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IrredundantSetCount"],{n,0,7}]

IsolatedPointCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"IsolatedPointCount"],{n,0,7}]

KatzCentralities

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"KatzCentralities"],{n,0,7}]

KirchhoffIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"KirchhoffIndex"],{n,0,7}]

KirchhoffSumIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"KirchhoffSumIndex"],{n,0,7}]

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LaplacianPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

LaplacianSpectralRadius

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LaplacianSpectralRadius"],{n,0,7}]

LaplacianSpectralRatio

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LaplacianSpectralRadius"],{n,0,7}]

LeafCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LeafCount"],{n,0,7}]

Likelihood

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Likelihood"],{n,0,7}]

LinearArboricity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LinearArboricity"],{n,0,7}]

LocalClusteringCoefficients

(Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LocalClusteringCoefficients"],{n,0,7}])//Column

LongestCycles

Do[Print[{
gn={"DorogovtsevGoltsevMendes",n},
Flatten@FindLongestCycle[GraphData@gn,"MaxLength"->3 2^(n-1)]//Timing}],{n,7}]

NextDGMLongestCycle[cyc_List]:=Module[{n=Log2[2 Length[cyc]/3]+1,g,m,pairs=Partition[cyc,2,1,1]},
m=AdjacencyList[g=DorogovtsevGoltsevMendesGraph[n]];
Flatten[{#,#3}&@@@(Flatten/@Transpose[{pairs,Flatten[Complement[Intersection@@m[[#]],cyc]&/@pairs]}])]
]

NestList[NextDGMLongestCycle,{1,2,3},4]

LongestCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LongestCycleCount"],{n,0,7}]

LongestPathCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LongestPathCount"],{n,0,7}]

LongestPathLength

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LongestPathLength"],{n,0,7}]

LovaszNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LovaszNumber"],{n,0,7}]

LowerCliqueNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LowerCliqueNumber"],{n,0,7}]

LowerIndependenceNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LowerIndependenceNumber"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>0]

LowerMatchingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"LowerMatchingNumber"],{n,0,7}]

MatchingGeneratingPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MatchingGeneratingPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

MatchingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MatchingNumber"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>1]

MatchingPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MatchingPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

MaximalCliqueCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalCliqueCount"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>1]

MaximalCliquePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalCliquePolynomial"],{n,0,7}]/.f_Function:>f[x])

MaximalIndependencePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalIndependencePolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalIndependentEdgeSetCount"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>0]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalIndependentVertexSetCount"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>0]

MaximalIrredundancePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalIrredundancePolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MaximalIrredundantSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalIrredundantSetCount"],{n,0,7}]

MaximalMatchingGeneratingPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalMatchingGeneratingPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MaximumCliqueCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumCliqueCount"],{n,0,7}]

FullSimplify[SeriesCoefficient[FindGeneratingFunction[seq,x],{x,0,n}],n>0]

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumIndependentEdgeSetCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumIndependentVertexSetCount"],{n,0,7}]

MaximumIrredundantSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumIrredundantSetCount"],{n,0,7}]

MaximumLeafNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumLeafNumber"],{n,0,7}]

MaximumVertexDegree

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximumVertexDegree"],{n,0,7}]

FindSequenceFunction[seq,n+1]

MeanClusteringCoefficient

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MeanClusteringCoefficient"],{n,0,7}]

Table[(4/5)(((6^(n-1))+3/2)/((2^(n-1))((3^(n-1))+1))),{n,0,7}]

Limit[(4/5)(((6^(n-1))+3/2)/((2^(n-1))((3^(n-1))+1))),n->∞]

https://arxiv.org/pdf/cond-mat/0112143.pdf

MeanDistance

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MeanDistance"],{n,0,7}]

FindSequenceFunction[seq,n+1]

Median

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Median"],{n,0,7}]

MetricDimension

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MetricDimension"],{n,0,7}]

MinimalConnectedDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalConnectedDominatingSetCount"],{n,0,7}]

MinimalConnectedDominationPolynomial

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalConnectedDominationPolynomial"],{n,0,7}]/.f_Function:>f[x]

MinimalDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalDominatingSetCount"],{n,0,7}]

MinimalDominationPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MaximalMatchingGeneratingPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimalEdgeCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalEdgeCoverCount"],{n,0,7}]

MinimalEdgeCoverPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalEdgeCoverPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimalEdgeCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalEdgeCutCount"],{n,0,7}]

MinimalEdgeCutPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalEdgeCutPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalTotalDominatingSetCount"],{n,0,7}]

MinimalTotalDominationPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalTotalDominationPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimalVertexCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalVertexCoverCount"],{n,0,7}]

MinimalVertexCoverPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalVertexCoverPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimalVertexCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalVertexCutCount"],{n,0,7}]

MinimalVertexCutPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimalVertexCutPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

MinimumCliqueCoveringCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumCliqueCoveringCount"],{n,0,7}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumConnectedDominatingSetCount"],{n,0,7}]

FindSequenceFunction[DeleteMissing@seq,n]

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumCoveringsByMaximalCliquesCount"],{n,0,7}]

MinimumCyclicEdgeCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumCyclicEdgeCutCount"],{n,0,7}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumDistinguishingLabelingCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

MinimumDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumDominatingSetCount"],{n,0,7}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumEdgeCoverCount"],{n,0,7}]

MinimumEdgeCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumEdgeCutCount"],{n,0,7}]

MinimumLeafNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumLeafNumber"],{n,0,7}]

MinimumPathCoveringCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumPathCoveringCount"],{n,0,7}]

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumTotalDominatingSetCount"],{n,0,7}]

MinimumVertexCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumVertexCoverCount"],{n,0,7}]

MinimumVertexCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumVertexCutCount"],{n,0,7}]

MinimumVertexDegree

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumVertexDegree"],{n,0,7}]

MolecularTopologicalIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MolecularTopologicalIndex"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

Nonhamiltonian

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Nonhamiltonian"],{n,0,7}]

NoPerfectMatching

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"NoPerfectMatching"],{n,0,7}]

OddChordlessCycleCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"OddChordlessCycleCount"],{n,0,7}]

OddChordlessCyclePolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"OddChordlessCyclePolynomial"],{n,0,7}]/.f_Function:>f[x])

Outerplanar

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Outerplanar"],{n,0,7}]

PageRankCentralities

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PageRankCentralities"],{n,0,7}]

PathCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PathCount"],{n,0,7}]

PathCoveringNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PathCoveringNumber"],{n,0,7}]

PathPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PathPolynomial"],{n,k0=0,kmax=7}]/.f_Function:>f[x])//Column

PentagonCount

A367967

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PentagonCount"],{n,0,7}]

FindGeneratingFunction[seq,x]

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CycleCount[DorogovtsevGoltsevMendesGraph[n],5]//Timing}],{n,0,10}];

seq={0,0,3,27,126,462,1521,4761,14556,44028,132543};

gf=FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

seq=Table[(3/4)((3^(n+1))-2n-4(n^2)-3),{n,0,27}]

StringLength[ToString[seq]]

FindLinearRecurrence[seq]

LinearRecurrence[{6,-12,10,-3},{0,0,3,27},20]

Cf. A003462(n) (3-cycles), A290764(n-1) (4-cycles), A367967 (5-cycles), XXX (6-cycles).

gf=FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

CoefficientList[Series[((3 (x^2) (1+3 x))/((((-1+x))^3) (-1+3 x))),{x,0,20}],x]

Perfect

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Perfect"],{n,0,7}]

PerfectMatching

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"PerfectMatching"],{n,0,7}]

Periphery

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Periphery"],{n,0,7}]

Planar

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Planar"],{n,0,7}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ProjectivePlaneCrossingNumber"],{n,0,7}]

Pseudoarboricity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Pseudoarboricity"],{n,0,7}]

Ptolemaic

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Ptolemaic"],{n,0,7}]

QChromaticPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"QChromaticPolynomial"],{n,k0=0,kmax=7}]/.f_Function:>f[x])//Column

QuadraticallyEmbeddable

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"QuadraticallyEmbeddable"],{n,0,7}]

QuadraticEmbeddingConstant

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"QuadraticEmbeddingConstant"],{n,0,7}]

RadialityCentralities

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RadialityCentralities"],{n,0,7}]

RadioLabelingCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RadioLabelingCount"],{n,0,7}]

RadioNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RadioNumber"],{n,0,7}]

Radius

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Radius"],{n,0,7}]

Rank

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Rank"],{n,0,7}]

RankPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RankPolynomial"],{n,k0=0,7}]/.f_Function:>f[x,y])//Column

coef=Factor/@FindLinearRecurrence[poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RectilinearCrossingNumber"],{n,0,7}]

Regular

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Regular"],{n,0,7}]

RegularParameters

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"RegularParameters"],{n,0,7}]//Column

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ReliabilityPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

ResistanceMatrix

MatrixForm/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ResistanceMatrix"],{n,6}]

ScrambleNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ScrambleNumber"],{n,0,7}]

ShannonCapacity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ShannonCapacity"],{n,0,7}]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SigmaPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Singlecross

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Singlecross"],{n,0,7}]

Skewness

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Skewness"],{n,0,7}]

SpanningTreeCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SpanningTreeCount"],{n,0,7}]

SpectralRadius

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SpectralRadius"],{n,0,7}]

SquareCount

A290764

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SquareCount"],{n,0,7}]

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CycleCount[DorogovtsevGoltsevMendesGraph[n],4]//Timing}],{n,0,10}];

seq={0,0,3,15,54,174,537,1629,4908,14748,44271};

gf=FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n≥0]

SquareFree

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SquareFree"],{n,0,7}]

StabilityIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"StabilityIndex"],{n,0,7}]

StarArboricity

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"StarArboricity"],{n,0,7}]

StatusCentralities

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"StatusCentralities"],{n,0,7}]

Strength

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Strength"],{n,0,7}]

StronglyPerfect

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"StronglyPerfect"],{n,0,7}]

StronglyRegular

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"StronglyRegular"],{n,0,7}]

Switchable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Switchable"],{n,0,7}]

Symmetric

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Symmetric"],{n,0,7}]

SymmetricallyDistinctVertexPairCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SymmetricallyDistinctVertexPairCount"],{n,0,7}]

FindGeneratingFunction[seq,x]//Head//Timing

SymmetricallyDistinctVertexPairSignature

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"SymmetricallyDistinctVertexPairSignature"],{n,0,7}]

Thickness

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Thickness"],{n,0,7}]

TopologicalIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TopologicalIndex"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

ToroidalCrossingNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"ToroidalCrossingNumber"],{n,0,7}]

TotalDominatingSetCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TotalDominatingSetCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

TotalDominationNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TotalDominationNumber"],{n,0,7}]

TotalDominationPolynomial

(poly=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TotalDominationPolynomial"],{n,0,7}]/.f_Function:>f[x])//Column

Toughness

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Toughness"],{n,0,7}]

Traceable

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Traceable"],{n,0,7}]

TreeDepth

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TreeDepth"],{n,0,7}]

TreeNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TreeNumber"],{n,0,7}]

Treewidth

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Treewidth"],{n,0,7}]

Triameter

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Triameter"],{n,0,7}]

FindSequenceFunction[Rest@seq,n]

TriangleCount

A003462

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TriangleCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

Do[Print[{gn={"DorogovtsevGoltsevMendes",n},CycleCount[DorogovtsevGoltsevMendesGraph[n],3]//Timing}],{n,0,10}];

seq={0,1,4,13,40,121,364,1093,3280,9841,29524};

gf=FindGeneratingFunction[seq,x]

SeriesCoefficient[gf,{x,0,n}]

TriangleFree

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TriangleFree"],{n,0,7}]

TuttePolynomial

(Table[GraphData[{"DorogovtsevGoltsevMendes",n},"TuttePolynomial"],{n,0,7}]/.f_Function:>f[x,y])//Column

Unicyclic

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Unicyclic"],{n,0,7}]

UniquelyColorable

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"UniquelyColorable"],{n,0,7}]

UniquelyPancyclic

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"UniquelyPancyclic"],{n,0,7}]

UnitDistance

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"UnitDistance"],{n,0,7}]

UpperDominationNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"UpperDominationNumber"],{n,0,7}]

UpperIrredundanceNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"UpperIrredundanceNumber"],{n,0,7}]

VertexConnectivity

Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexConnectivity"],{n,0,7}]

VertexCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

VertexCoverCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexCoverCount"],{n,0,7}]

FindSequenceFunction[seq,n+1]

x FindGeneratingFunction[DeleteMissing@seq,x]

VertexCoverNumber

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexCoverNumber"],{n,0,7}]

FindSequenceFunction[seq,n+1]

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexCoverPolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

VertexCutCount

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexCutCount"],{n,0,7}]

VertexDegreePolynomial

(poly=Factor@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexDegreePolynomial"],{n,k0=0,7}]/.f_Function:>f[x])//Column

Table[2(x^(2^n))+Sum[(3^(n-k))(x^(2^k)),{k,n}],{n,0,7}]//Column

2(x^(2^n))+Sum[(3^(n-k))(x^(2^k)),{k,n}]

https://arxiv.org/pdf/cond-mat/0112143.pdf

VertexDegrees

Tally/@Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexDegrees"],{n,0,7}]//Column

VertexTransitive

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"VertexTransitive"],{n,0,7}]

WeisfeilerLemanDimension

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"WeisfeilerLemanDimension"],{n,0,7}]

FindSequenceFunction[seq,n+1]

WienerIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"WienerIndex"],{n,0,7}]

FindSequenceFunction[seq,n+1]//Head//Timing

WienerSumIndex

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"WienerSumIndex"],{n,0,7}]

FindSequenceFunction[seq,n+1]

Algorithms

Python

https://www.wolframcloud.com/obj/arnoudbuzing/DeployedResources/Function/DorogovtsevGoltsevMendesGraph/

DorogovtsevGoltsevMendesGraphPython[n_Integer]:=Module[{session,ba},
session=SelectFirst[
ExternalSessions[],
#["ID"]==="networkx"&,
StartExternalSession[{
"Python",
"ID"->"networkx",
"Evaluator"-><|
"Dependencies"->{"networkx==3.2.1"},
"EnvironmentName"->"networkx",
"PythonRuntime"->"3.11"|>,
"SessionProlog"->"import networkx"}]
];
ba=ExternalFunction[session,"lambda n: networkx.to_graph6_bytes(networkx.dorogovtsev_goltsev_mendes_graph(n))"][n];
ImportByteArray[ba,"Graph6"]
]

gs=Table[DorogovtsevGoltsevMendesGraphPython[n],{n,0,4}]

RecognizeGraph/@gs

VertexCount/@gs

Table[3((3^n)+1)/2,{n,0,10}]

EdgeCount/@gs

Table[(3^n),{n,4}]

Graphs.m

Table[DorogovtsevGoltsevMendesGraph[n],{n,0,4}]

IsomorphicGraphQ@@@Table[{DorogovtsevGoltsevMendesGraphPython[n],DorogovtsevGoltsevMendesGraph[n]},{n,0,5}]

GraphData["K3","Edges","List"]

Max[%]

GraphData

GraphDataClassString

GraphDataClassString["DorogovtsevGoltsevMendes"]

GraphDataString

file=OpenWrite["~/dgm.m"]

Do[
WriteString[file,GraphDataString[DorogovtsevGoltsevMendesGraph[n],
"StandardName"->{"DorogovtsevGoltsevMendes",n},
"Name"->ToString[n]<>"\[Hyphen]Dorogovtsev\[Hyphen]Goltsev\[Hyphen]Mendes graph",
"Classes"->{"DorogovtsevGoltsevMendes"},
"NotationRules"->{"DorogovtsevGoltsevMendes"->n},
"Information"->"Dorogovtsev-Goltsev-MendesGraph",
TimeConstraint->60,
"Skip"->{"ABCEnergy","ABCSpectralRadius","Arboricity","ConnectedInducedSubgraphPolynomial","ConnectedInducedSubgraphs","ConstructionNumber","CyclicEdgeConnectivity","DetourMatrix","EdgeCovers","GracefulLabeling","GracefulLabelings","IrredundacePolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSets","MaximumIndependentEdgeSets","MaximumIndependentVertexSets","MetricDimension","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","Minors","QuadraticEmbeddingConstant","RandicEnergy","SomborSpectralRadius","TotalDominationPolynomial","TuttePolynomial"}
]],
{n,5,7}]