Antelope Graph

Author

Eric W. Weisstein
June 28, 2024

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

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

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

Graphics

Graphs.m

GraphicsGrid[Partition[Table[AntelopeGraph[n,n]//StyleGraphs,{n,12}],UpTo[4]],Dividers->All,ImageSize->500]

GraphicsGrid[Partition[Table[GraphicsRow[ConnectedGraphComponents@AntelopeGraph[n,n]//StyleGraphs],{n,12}],UpTo[4]],Dividers->All,ImageSize->500]

GraphData

GraphicsRow[Table[GraphData[{"Antelope",{n,n}},"Graph","Grid"][[1]]//StyleGraphs,{n,7}],Dividers->All,ImageSize->500]

Special Cases

With[{cl="Antelope"},TextGrid[Sort[{cl/.GraphData[#,"NotationRules"],GraphData[#,"Name"]}&/@DeleteCases[GraphData[cl],{cl,_}]],Dividers->All]]

Standard names

Grid[Sort[{
#,
With[{cc=GraphData[#,"ConnectedComponents","Name"]},
Check[
StringJoin@@Riffle[Reverse@SortBy[Reverse/@Tally[cc],GraphData[#[[2]],"V"]&]/.{
{n_,"SingletonGraph"}:>ToString[n]<>"K1",
{n_,{"Path",k_}}:>ToString[n]<>"P"<>ToString[k],
{n_,{"Cycle",k_}}:>If[n==1,"",ToString[n]]<>"C"<>ToString[k]
},"+"],
cc]//Quiet
]
}&/@GraphData[{"Antelope","Disconnected","Nonempty"}]],Dividers->All,Alignment->Left]

Embeddings

<<MathWorld`Graphs`

Slideview

With[{cl="Antelope"},SlideView[Sort@{Column[Union@{#,{cl,cl/.GraphData[#,"NotationRules"]}},Alignment->Center],GetGraphData[#,"File"],Grid[Transpose[{GraphData[#,"Graph","All"],GraphData[#,"EmbeddingClasses"]}],Dividers->All]}&/@Sort@GraphData@cl]]

Toroidal

GraphData[#,"Image2D"]&/@GraphData[{"AntelopesTour","Connected"}]

GraphData[#,"Image3D"]&/@GraphData[{"AntelopesTour","Connected"}]

{"Antelope", {1, 1}} == "SingletonGraph"

Names

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

{"Antelope", {2, 2}} == {"Empty", 4}

Names

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

{"Antelope", {3, 3}} == {"Empty", 9}

Names

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

{"Antelope", {4, 4}} == {"Empty", 16}

Names

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

{"Antelope", {5, 5}} == "4P3+13K1"

Names

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

{"Antelope", {6, 6}} == "4P5+4P3+4K1"

Names

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

{"Antelope", {6, 9}}

GraphData[{"Antelope",{6,9}},"StandardName"]

All

GraphData[{"Antelope",{6,9}},"Graph","All"]//StyleGraphs

Halin

GraphData[{"Antelope",{6,9}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"Antelope",{6,9}},"LCFSignature"]

GraphData[{"Antelope",{6,9}},"Graph","LCF"]//StyleGraphs

UnitDistance

GraphData[{"Antelope",{6,9}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{6,9}}],Debug->True]//Timing

GraphData[{"Antelope",{6,9}},"Graph","UnitDistance"]//StyleGraphs

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Antelope",{6,9}},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"Antelope",{6,9}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"Antelope",{6,9}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{6,9}}],TimeConstraint->300,"Results"->All]//Timing}

FindSubgraphMatchstickEmbedding[GraphData[{"Antelope",{6,9}}]]

gs=UniqueEmbeddings[l]

Matchstick FindSubgraphMatchstickEmbedding

FindSubgraphMatchstickEmbedding[GraphData[{"Antelope",{6,9}}]]

Matchstick construction

GraphData[{"Antelope",{6,9}},"Graph","MinimalCrossing"][[1]]

2-First[v[[45]]-v[[40]]]

e[l__]:=Sequence@@Partition[{l},2,1]
v=Join[CirclePoints[{Csc[Pi/8]/2,0},8],CirclePoints[{9-3+Csc[(π/8)],0},{Csc[Pi/8]/2,0},8],CirclePoints[{19-9+(Sqrt[2])+(1/2) Csc[(π/14)]+(3/2) Csc[(π/8)],0},Csc[Pi/14]/2,14]];
v=Join[v,
Table[v[[5]]-{i,0},{i,2}],Table[v[[1]]+{i,0},{i,2}],
(v[[3]]+Through[{Cos,Sin}[Pi/2+# Pi/4]])&/@{-1,1},
Table[v[[13]]-{i,0},{i,2}],Table[v[[9]]+{i,0},{i,2}],
(v[[11]]+Through[{Cos,Sin}[Pi/2+# Pi/4]])&/@{-1,1},
Flatten[Table[v[[27]]+j Through[{Cos,Sin}[Pi+i Pi/4]],{j,2},{i,{-1,1}}],1],
Flatten[Table[v[[20]]+j Through[{Cos,Sin}[i Pi/4]],{j,2},{i,{-1,1}}],1],
{v[[22]]+Through[{Cos,Sin}[Pi/4]],v[[25]]+Through[{Cos,Sin}[3Pi/4]],v[[29]]+Through[{Cos,Sin}[5Pi/4]],v[[18]]+Through[{Cos,Sin}[7Pi/4]]}
];

v=FullSimplify@Map[ToAlgebraicRoot,v,{2}];//Timing

g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,7,8,1,33,34],e[36,3,35],e[32,31,5],
e[9,10,11,12,13,14,15,16,9,39,40],e[38,37,13],e[42,11,41],e[20,21,22,23,24,25,26,27,28,29,30,17,18,19,20],e[50,48,20,47,49],e[45,43,27,44,46],{22,51},{25,52},{29,53},{18,54}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

EmbeddingClasses[g]//Quiet//Timing

Matchstick

GraphData[{"Antelope",{6,9}},"Graph","Matchstick"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{6,9}},{"RCN","CN"}]

GraphData[{"Antelope",{6,9}},"Graph","MinimalCrossing"]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"Antelope",{6,9}},"Edges"]

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

RecognizeGraph[g]

MinimalCrossing construction

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{6,9}}],"CrossingNumber"->-1]]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

3D default

Graph3D[GraphData[{"Antelope",{6,9}},"Edges"]]

3D All

GraphData[{"Antelope",{6,9}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{6,9}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{6,9}},"Polyhedral"]

GraphData[{"Antelope",{6,9}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {7,7}} == "C24+C16+C8+K1"

Names

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

{"Antelope", {7, 9}}

Names

GraphData[{"Antelope",{7,9}},"StandardName"]

GraphData[{"Antelope",{7,9}},"StandardNames"]

Primary

GraphData[{"Antelope",{7,9}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"Antelope",{7,9}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{7,9}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{7,9}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{7,9}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{7,9}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{7,9}},"Graph","Degenerate"]

Configuration

GraphData[{"Antelope",{7,9}},"Configuration"]

GraphData[{"Antelope",{7,9}},"Graph","Configuration"]//StyleGraphs

Perspective

GraphData[{"Antelope",{7,9}},"Graph","Perspective"]//StyleGraphs

HoneycombToroidal construction

GraphData[{"Antelope",{7,9}},"NotationRules","HoneycombToroidal"]

GeneralizedPetersen

GraphData[{"Antelope",{7,9}},"Graph","GeneralizedPetersen"]//StyleGraphs

IGraph

GraphData[{"Antelope",{7,9}},"Graph","IGraph"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{7,9}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{7,9}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{7,9}},"Hamiltonian"]

GraphData[{"Antelope",{7,9}},"HamiltonianCycleCount"]

Planar

GraphData[{"Antelope",{7,9}},"Planar"]

GraphData[{"Antelope",{7,9}},"Graph","Planar"]//StyleGraphs

MinimalLocalCrossing

GraphData[{"Antelope",{7,9}},"LocalCrossingNumber"]

GraphData[{"Antelope",{7,9}},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{7,9}},{"RCN","CN"}]

GraphData[{"Antelope",{7,9}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->5 3600])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{7,9}}],TimeConstraint->86400])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{7,9}}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{7,9}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{7,9}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{7,9}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union//Timing

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot[MinimalCrossingGraph[GraphData[{"Antelope",{7,9}}],VertexLabels->None,"CrossingNumber"->2]]

Mean[v[[{2,58}]]]//FullSimplify

{-(1/14) Cot[(π/14)] (1+6 Sin[(π/14)]),(1/32) (1+15 Cos[(π/7)] Csc[(π/14)])}//N

v={{0,(1/2) Csc[(π/14)]},{-(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]},{-(1/2) Cos[((3 π)/14)] Csc[(π/14)],(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{-(1/2) Cot[(π/14)],(1/2)},{-(1/2) Cot[(π/14)],-(1/2)},{-(1/2) Cos[((3 π)/14)] Csc[(π/14)],-(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{-(1/2) Csc[(π/14)] Sin[(π/7)],-(1/2) Cos[(π/7)] Csc[(π/14)]},{0,-(1/2) Csc[(π/14)]},{(1/2) Csc[(π/14)] Sin[(π/7)],-(1/2) Cos[(π/7)] Csc[(π/14)]},{(1/2) Cos[((3 π)/14)] Csc[(π/14)],-(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{(1/2) Cot[(π/14)],-(1/2)},{(1/2) Cot[(π/14)],(1/2)},{(1/2) Cos[((3 π)/14)] Csc[(π/14)],(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]},{-(1/4) Cos[((3 π)/14)] Csc[(π/14)],-(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{0,-(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{(1/4) Cos[((3 π)/14)] Csc[(π/14)],-(1/2) Csc[(π/14)] Sin[((3 π)/14)]},{-(3/8) Cot[(π/14)],-(1/2)},{-(1/4) Cot[(π/14)],-(1/2)},{-(1/8) Cot[(π/14)],-(1/2)},{0,-(1/2)},{(1/8) Cot[(π/14)],-(1/2)},{(1/4) Cot[(π/14)],-(1/2)},{(3/8) Cot[(π/14)],-(1/2)},{-(1/8) Cot[(π/14)],(1/2) (-(1/2)-(1/2) Csc[(π/14)] Sin[((3 π)/14)])},{(1/8) Cot[(π/14)],(1/2) (-(1/2)-(1/2) Csc[(π/14)] Sin[((3 π)/14)])},{-(1/3) Cot[(π/14)],(1/2)},{-(1/6) Cot[(π/14)],(1/2)},{0,(1/2)},{(1/6) Cot[(π/14)],(1/2)},{(1/3) Cot[(π/14)],(1/2)},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]+(1/2) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]-(3/4) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) Cos[(π/7)] Csc[(π/14)]-(7/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]+(1/4) (-(1/6) Cot[(π/14)]-(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(7/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]+(1/2) (-(1/6) Cot[(π/14)]-(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(7/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(3/4) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(7/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]+(1/7) (-(1/6) Cot[(π/14)]-(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(2/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(4/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(5/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(6/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{-(1/6) Cot[(π/14)],(1/2) Cos[(π/7)] Csc[(π/14)]-(5/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{-(7/5),(7/5)},{-(1/14) Cot[(π/14)] (1+6 Sin[(π/14)]),2},{(1/2) Csc[(π/14)] Sin[(π/7)]+(1/7) (-(1/6) Cot[(π/14)]-(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(2/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(4/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(5/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{(1/2) Csc[(π/14)] Sin[(π/7)]-(6/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{-(1/6) Cot[(π/14)],(1/2) Cos[(π/7)] Csc[(π/14)]-(3/8) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},{-(1/6) Cot[(π/14)],(1/2) Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])},{-(1/6) Cot[(π/14)]+(1/4) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]-(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)])),(1/2) Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/4) ((1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/4) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)]))},{-(1/6) Cot[(π/14)]+(1/2) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]-(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)])),(1/2) Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/2) ((1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/4) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)]))},{-(1/6) Cot[(π/14)]-(3/4) (-(1/6) Cot[(π/14)]-(1/2) Csc[(π/14)] Sin[(π/7)]+(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)])),(1/2) Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])-(3/4) ((1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/4) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)]))},{(1/2) Csc[(π/14)] Sin[(π/7)]-(3/7) ((1/6) Cot[(π/14)]+(1/2) Csc[(π/14)] Sin[(π/7)]),(1/2) Cos[(π/7)] Csc[(π/14)]+(1/2) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)])+(1/4) (-(1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])},
{(1/2) Csc[(π/14)] Sin[(π/7)],(1/2) (Cos[(π/7)] Csc[(π/14)]+(1/4) ((1/2)-(1/2) Cos[(π/7)] Csc[(π/14)]))},
{-(1/6) Cot[(π/14)],(1/2) ((1/2)+(1/2) Cos[(π/7)] Csc[(π/14)])}};

Mean[v[[{32,14}]]]

e=Join[
Partition[Range[14],2,1,1],
Partition[{6,15,16,17,10},2,1],
Partition[Range[18,24],2,1],
Partition[Range[27,31],2,1],
Partition[Range[32,36],2,1],
Partition[Range[37,40],2,1],
{{19,25},{25,16},{16,26},{26,23},{4,18},{12,24},{4,27},{12,31},{28,47},{30,37},{39,36},{35,41},{41,42},{42,43},{47,46},{46,45},{45,44},{40,48},{48,49},{50,51},{52,53},{35,50},{51,52},{55,54},{48,56},{55,56},{53,43},{44,53},{48,57},{2,58},{49,33},{53,61},{60,61},{59,60},{58,59},{55,57},{54,60},{14,62},{32,62},{55,63},{63,45}}
];

g=Graph[Range[Length[v]],UndirectedEdge@@@e,VertexCoordinates->v,VertexLabels->"Name"]

GraphData[{"Antelope",{7,9}},{"V","M"}]

Tally[GraphData[{"Antelope",{7,9}},"VertexDegrees"]]

GraphData[{"Antelope",{7,9}},"Connected"]

Through[{VertexCount,EdgeCount}[g]]

Tally[VertexDegree[g]]

RecognizeGraph[g]

GraphPlot@RotateGraph[,{58,4}]

Circulant

GraphData[{"Antelope",{7,9}},"Circulant"]

GraphData[{"Antelope",{7,9}},"Graph","Circulant"]//StyleGraphs

Complement graph

compl=GraphData[{"Antelope",{7,9}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{7,9}},"Line"]

LineGraphQ[GraphData[{"Antelope",{7,9}}]]//Timing

RootGraph

roots=Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{7,9}}&]

RootGraph[GraphData[{"Antelope",{7,9}}]]

RecognizeGraph[%]

Halin

GraphData[{"Antelope",{7,9}},"Halin"]

GraphData[{"Antelope",{7,9}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{7,9}},"Rigid"]

GraphData[{"Antelope",{7,9}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{7,9}},"Flexible"]

GraphData[{"Antelope",{7,9}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{7,9}},"LCF"]

GraphData[{"Antelope",{7,9}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{7,9}},"LCFSignature"]

GraphData[{"Antelope",{7,9}},"LCFNotations"]

GraphData[{"Antelope",{7,9}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{7,9}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{7,9}}],Debug->True]//Timing

GraphData[{"Antelope",{7,9}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{7,9}},"VertexList"],GraphData[{"Antelope",{7,9}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{7,9}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{7,9}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{7,9}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{7,9}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{7,9}}],TimeConstraint->300,"Results"->All]//Timing}

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

gs={,};

Matchstick

GraphData[{"Antelope",{7,9}},"Matchstick"]

GraphData[{"Antelope",{7,9}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{7,9}},"Edges"]]

3D All

GraphData[{"Antelope",{7,9}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{7,9}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{7,9}},"Polyhedral"]

GraphData[{"Antelope",{7,9}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {8, 8}}

Names

GraphData[{"Antelope",{8,8}},"StandardName"]

GraphData[{"Antelope",{8,8}},"StandardNames"]

Primary

GraphData[{"Antelope",{8,8}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{8,8}},"Configuration"]

GraphData[{"Antelope",{8,8}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{8,8}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{8,8}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{8,8}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{8,8}},"EdgeList"]]

Default embedding

GraphPlot[GraphData[{"Antelope",{8,8}},"Edges"],GraphLayout->"SpringEmbedding"]

dx=1/6;
v=CirclePoints[30];
v=Join[v,Subdivide[v[[30]],v[[16]],7][[2;;-2]],Subdivide[v[[1]],v[[15]],7][[2;;-2]]];
newv=Join[v[[41]]+{# dx,0}&/@{2,3},v[[40]]+{# dx,0}&/@{1,3,4}];
v=Join[v,Join@@Function[f,f #&/@newv]/@Tuples[{-1,1},2]];
v=Join[v,{v[[8]]-{dx,0},v[[23]]+{dx,0}}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[7,63,9],e[22,64,24],e[{1,15}],e[{16,30}],e[1,37,38,53,61,57,7],e[15,42,41,58,56,62,9],e[30,31,32,43,51,47,24],e[16,36,35,48,46,52,22],e[32,39,60,13],e[38,33,50,18],e[35,40,55,3],e[41,34,45,28],e[20,49,51],e[26,44,46],e[61,59,11],e[5,54,56],{33,45},{34,50},{40,60},{39,55}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{8,8}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{8,8}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{8,8}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{8,8}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{8,8}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{8,8}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{8,8}},"Planar"]

GraphData[{"Antelope",{8,8}},"Graph","Planar"]//StyleGraphs

MinimalLocalCrossing

GraphData[{"Antelope",{8,8}},"LocalCrossingNumber"]

GraphData[{"Antelope",{8,8}},"Graph","MinimalLocalCrossing"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{8,8}},{"RCN","CN"}]

GraphData[{"Antelope",{8,8}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{8,8}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{8,8}}],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{8,8}}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{8,8}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{8,8}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{8,8}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

GraphPlot@RotateGraph[,-.7]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot@RotateGraph[,-.8]

v={#/10,#2/14}&@@@{{0,0},{5,0},{10,0},{0,1},{1,1},{3,1},{4,1},{5,1},{6,1},{7,1},{9,1},{10,1},{3,2},{4,2},{6,2},{7,2},{3,3},{4,3},{5,3},{6,3},{7,3},{0,4},{1,4},{3,4},{7,4},{9,4},{10,4},{0,5},{1,5},{3,5},{4,5},{5,5},{6,5},{7,5},{9,5},{10,5},{0,8},{1,8},{2,8},{3,8},{4,8},{6,8},{7,8},{8,8},{9,8},{10,8},{4,9},{6,9},{4,10},{5,10},{6,10},{4,11},{6,11},{4,12},{5,11},{6,12},{4,13},{6,13},{0,14},{4,14},{5,14},{6,14},{10,14},{5,13}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,12,27,36,46,63,62,61,60,59,37,28,22,4,1],e[{4,12}],e[37,38,39,40,41,47,49,52,54,57,60],e[62,58,56,53,51,48,42,43,44,45,46],e[54,64,56,55,54],e[49,50,51],e[{28,36}],e[22,23,42],e[41,26,27],{23,29},{26,35},e[6,13,17,24,34],e[10,16,21,25,30],e[8,14,19,15,8],e[{17,21}]},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

MinimalCrossingGraph[GraphData[{"Antelope",{8,8}}],"CrossingNumber"->4,RandomSeed->1234]

GraphPlot@RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{8,8}}],"CrossingNumber"->4,RandomSeed->1234],{52,34}]

v=Append[{#,If[#2>7,#2+1,#2]}&@@@{{0,0},{5,0},{10,0},{0,1},{1,1},{2,1},{4,1},{5,1},{6,1},{8,1},{9,1},{10,1},{4,2},{6,2},{2,3},{3,3},{4,3},{5,3},{6,3},{7,3},{8,3},{3,4},{7,4},{0,5},{1,5},{3,5},{4,5},{5,5},{6,5},{7,5},{8,5},{9,5},{10,5},{0,6},{1,6},{3,6},{4,6},{6,6},{7,6},{3,7},{7,7},{10,7},{3,8},{4,8},{5,8},{6,8},{7,8},{3,9},{4,9},{6,9},{7,9},{3,10},{4,10},{5,10},{6,10},{7,10},{3,11},{7,11},{0,12},{10,12},{0,13},{5,13},{10,13}},{7,8}];
g=RotateGraph[Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,12,33,42,60,63,62,61,59,34,24,4,1],e[4,5,6,7,8,9,10,11,12],e[8,13,18,14,8],e[6,15,16,17,18,19,20,21,10],e[16,22,30],e[20,23,26,35,34],e[24,25,36,37,38,39,41,64,47,51,56,58,59],e[36,40,43,48,52,57,60],e[26,27,28,29,30,31,32,33],e[31,42],{32,41},e[45,49,54,50,45],e[43,44,45,46,47],e[52,53,54,55,56],{25,35}},VertexCoordinates->v,VertexLabels->Automatic],Pi/2]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{8,8}},"Circulant"]

GraphData[{"Antelope",{8,8}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{8,8}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{8,8}},"Line"]

LineGraphQ[GraphData[{"Antelope",{8,8}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{8,8}}&]

RootGraph[GraphData[{"Antelope",{8,8}}]]

Halin

GraphData[{"Antelope",{8,8}},"Halin"]

GraphData[{"Antelope",{8,8}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{8,8}},"Rigid"]

GraphData[{"Antelope",{8,8}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{8,8}},"Flexible"]

GraphData[{"Antelope",{8,8}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{8,8}},"LCF"]

GraphData[{"Antelope",{8,8}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{8,8}},"LCFSignature"]

GraphData[{"Antelope",{8,8}},"LCFNotations"]

GraphData[{"Antelope",{8,8}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{8,8}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{8,8}}],Debug->True]//Timing

GraphData[{"Antelope",{8,8}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{8,8}},"VertexList"],GraphData[{"Antelope",{8,8}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{8,8}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{8,8}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{8,8}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{8,8}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{8,8}}],TimeConstraint->300,"Results"->All]//Timing}

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

g=;

Matchstick

GraphData[{"Antelope",{8,8}},"Matchstick"]

GraphData[{"Antelope",{8,8}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{8,8}},"Edges"]]

3D All

GraphData[{"Antelope",{8,8}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{8,8}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{8,8}},"Polyhedral"]

GraphData[{"Antelope",{8,8}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {9, 9}}

Names

GraphData[{"Antelope",{9,9}},"StandardName"]

GraphData[{"Antelope",{9,9}},"StandardNames"]

Primary

GraphData[{"Antelope",{9,9}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{9,9}},"Configuration"]

GraphData[{"Antelope",{9,9}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{9,9}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{9,9}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{9,9}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{9,9}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{9,9}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{9,9}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{9,9}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{9,9}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{9,9}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{9,9}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{9,9}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{9,9}},"Planar"]

GraphData[{"Antelope",{9,9}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{9,9}},{"RCN","CN"}]

GraphData[{"Antelope",{9,9}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{9,9}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{9,9}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{9,9}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{9,9}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{9,9}}],"CrossingNumber"->cn])//PolyhedralGraphQ

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{9,9}}],"CrossingNumber"->cn,RandomSeed->n]],{n,250}],Max[OuterFace[#]]<=GraphData[{"Antelope",{9,9}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{9,9}}],"CrossingNumber"->cn,RandomSeed->i],{i,250}]//Union//Timing

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{9,9}},"Circulant"]

GraphData[{"Antelope",{9,9}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{9,9}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{9,9}},"Line"]

LineGraphQ[GraphData[{"Antelope",{9,9}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{9,9}}&]

RootGraph[GraphData[{"Antelope",{9,9}}]]

Halin

GraphData[{"Antelope",{9,9}},"Halin"]

GraphData[{"Antelope",{9,9}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{9,9}},"Rigid"]

GraphData[{"Antelope",{9,9}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{9,9}},"Flexible"]

GraphData[{"Antelope",{9,9}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{9,9}},"LCF"]

GraphData[{"Antelope",{9,9}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{9,9}},"LCFSignature"]

GraphData[{"Antelope",{9,9}},"LCFNotations"]

GraphData[{"Antelope",{9,9}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{9,9}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{9,9}}],Debug->True]//Timing

GraphData[{"Antelope",{9,9}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{9,9}},"VertexList"],GraphData[{"Antelope",{9,9}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{9,9}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{9,9}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{9,9}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{9,9}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{9,9}}],TimeConstraint->300,"Results"->All]//Timing}

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

g=;

Matchstick

GraphData[{"Antelope",{9,9}},"Matchstick"]

GraphData[{"Antelope",{9,9}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{9,9}},"Edges"]]

3D All

GraphData[{"Antelope",{9,9}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{9,9}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{9,9}},"Polyhedral"]

GraphData[{"Antelope",{9,9}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {10, 10}}

Names

GraphData[{"Antelope",{10,10}},"StandardName"]

GraphData[{"Antelope",{10,10}},"StandardNames"]

Primary

GraphData[{"Antelope",{10,10}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{10,10}},"Configuration"]

GraphData[{"Antelope",{10,10}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{10,10}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{10,10}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{10,10}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{10,10}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{10,10}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{10,10}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{10,10}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{10,10}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{10,10}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{10,10}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{10,10}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{10,10}},"Planar"]

GraphData[{"Antelope",{10,10}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{10,10}},{"RCN","CN"}]

GraphData[{"Antelope",{10,10}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{10,10}}],TimeConstraint->1])//AbsoluteTiming

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

(cn=CrossingNumber[GraphData[{"Antelope",{10,10}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{10,10}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{10,10}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{10,10}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{10,10}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{10,10}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{10,10}},"Circulant"]

GraphData[{"Antelope",{10,10}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{10,10}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{10,10}},"Line"]

LineGraphQ[GraphData[{"Antelope",{10,10}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{10,10}}&]

RootGraph[GraphData[{"Antelope",{10,10}}]]

Halin

GraphData[{"Antelope",{10,10}},"Halin"]

GraphData[{"Antelope",{10,10}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{10,10}},"Rigid"]

GraphData[{"Antelope",{10,10}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{10,10}},"Flexible"]

GraphData[{"Antelope",{10,10}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{10,10}},"LCF"]

GraphData[{"Antelope",{10,10}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{10,10}},"LCFSignature"]

GraphData[{"Antelope",{10,10}},"LCFNotations"]

GraphData[{"Antelope",{10,10}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{10,10}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{10,10}}],Debug->True]//Timing

GraphData[{"Antelope",{10,10}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{10,10}},"VertexList"],GraphData[{"Antelope",{10,10}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{10,10}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{10,10}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{10,10}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{10,10}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{10,10}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{10,10}},"Matchstick"]

GraphData[{"Antelope",{10,10}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{10,10}},"Edges"]]

3D All

GraphData[{"Antelope",{10,10}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{10,10}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{10,10}},"Polyhedral"]

GraphData[{"Antelope",{10,10}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {11, 11}}

Names

GraphData[{"Antelope",{11,11}},"StandardName"]

GraphData[{"Antelope",{11,11}},"StandardNames"]

Primary

GraphData[{"Antelope",{11,11}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{11,11}},"Configuration"]

GraphData[{"Antelope",{11,11}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{11,11}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{11,11}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{11,11}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{11,11}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{11,11}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{11,11}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{11,11}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{11,11}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{11,11}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{11,11}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{11,11}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{11,11}},"Planar"]

GraphData[{"Antelope",{11,11}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{11,11}},{"RCN","CN"}]

GraphData[{"Antelope",{11,11}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{11,11}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{11,11}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{11,11}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{11,11}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{11,11}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{11,11}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{11,11}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{11,11}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{11,11}},"Circulant"]

GraphData[{"Antelope",{11,11}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{11,11}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{11,11}},"Line"]

LineGraphQ[GraphData[{"Antelope",{11,11}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{11,11}}&]

RootGraph[GraphData[{"Antelope",{11,11}}]]

Halin

GraphData[{"Antelope",{11,11}},"Halin"]

GraphData[{"Antelope",{11,11}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{11,11}},"Rigid"]

GraphData[{"Antelope",{11,11}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{11,11}},"Flexible"]

GraphData[{"Antelope",{11,11}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{11,11}},"LCF"]

GraphData[{"Antelope",{11,11}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{11,11}},"LCFSignature"]

GraphData[{"Antelope",{11,11}},"LCFNotations"]

GraphData[{"Antelope",{11,11}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{11,11}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{11,11}}],Debug->True]//Timing

GraphData[{"Antelope",{11,11}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{11,11}},"VertexList"],GraphData[{"Antelope",{11,11}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{11,11}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{11,11}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{11,11}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{11,11}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{11,11}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{11,11}},"Matchstick"]

GraphData[{"Antelope",{11,11}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{11,11}},"Edges"]]

3D All

GraphData[{"Antelope",{11,11}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{11,11}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{11,11}},"Polyhedral"]

GraphData[{"Antelope",{11,11}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {12, 12}}

Names

GraphData[{"Antelope",{12,12}},"StandardName"]

GraphData[{"Antelope",{12,12}},"StandardNames"]

Primary

GraphData[{"Antelope",{12,12}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{12,12}},"Configuration"]

GraphData[{"Antelope",{12,12}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{12,12}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{12,12}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{12,12}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{12,12}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{12,12}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{12,12}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{12,12}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{12,12}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{12,12}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{12,12}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{12,12}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{12,12}},"Planar"]

GraphData[{"Antelope",{12,12}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{12,12}},{"RCN","CN"}]

GraphData[{"Antelope",{12,12}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{12,12}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{12,12}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{12,12}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{12,12}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{12,12}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{12,12}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{12,12}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{12,12}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{12,12}},"Circulant"]

GraphData[{"Antelope",{12,12}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{12,12}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{12,12}},"Line"]

LineGraphQ[GraphData[{"Antelope",{12,12}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{12,12}}&]

RootGraph[GraphData[{"Antelope",{12,12}}]]

Halin

GraphData[{"Antelope",{12,12}},"Halin"]

GraphData[{"Antelope",{12,12}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{12,12}},"Rigid"]

GraphData[{"Antelope",{12,12}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{12,12}},"Flexible"]

GraphData[{"Antelope",{12,12}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{12,12}},"LCF"]

GraphData[{"Antelope",{12,12}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{12,12}},"LCFSignature"]

GraphData[{"Antelope",{12,12}},"LCFNotations"]

GraphData[{"Antelope",{12,12}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{12,12}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{12,12}}],Debug->True]//Timing

GraphData[{"Antelope",{12,12}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{12,12}},"VertexList"],GraphData[{"Antelope",{12,12}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{12,12}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{12,12}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{12,12}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{12,12}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{12,12}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{12,12}},"Matchstick"]

GraphData[{"Antelope",{12,12}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{12,12}},"Edges"]]

3D All

GraphData[{"Antelope",{12,12}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{12,12}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{12,12}},"Polyhedral"]

GraphData[{"Antelope",{12,12}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {13, 13}}

Names

GraphData[{"Antelope",{13,13}},"StandardName"]

GraphData[{"Antelope",{13,13}},"StandardNames"]

Primary

GraphData[{"Antelope",{13,13}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{13,13}},"Configuration"]

GraphData[{"Antelope",{13,13}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{13,13}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{13,13}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{13,13}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{13,13}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{13,13}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{13,13}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{13,13}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{13,13}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{13,13}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{13,13}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{13,13}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{13,13}},"Planar"]

GraphData[{"Antelope",{13,13}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{13,13}},{"RCN","CN"}]

GraphData[{"Antelope",{13,13}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{13,13}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{13,13}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{13,13}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{13,13}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{13,13}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{13,13}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{13,13}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{13,13}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{13,13}},"Circulant"]

GraphData[{"Antelope",{13,13}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{13,13}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{13,13}},"Line"]

LineGraphQ[GraphData[{"Antelope",{13,13}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{13,13}}&]

RootGraph[GraphData[{"Antelope",{13,13}}]]

Halin

GraphData[{"Antelope",{13,13}},"Halin"]

GraphData[{"Antelope",{13,13}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{13,13}},"Rigid"]

GraphData[{"Antelope",{13,13}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{13,13}},"Flexible"]

GraphData[{"Antelope",{13,13}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{13,13}},"LCF"]

GraphData[{"Antelope",{13,13}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{13,13}},"LCFSignature"]

GraphData[{"Antelope",{13,13}},"LCFNotations"]

GraphData[{"Antelope",{13,13}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{13,13}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{13,13}}],Debug->True]//Timing

GraphData[{"Antelope",{13,13}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{13,13}},"VertexList"],GraphData[{"Antelope",{13,13}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{13,13}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{13,13}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{13,13}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{13,13}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{13,13}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{13,13}},"Matchstick"]

GraphData[{"Antelope",{13,13}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{13,13}},"Edges"]]

3D All

GraphData[{"Antelope",{13,13}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{13,13}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{13,13}},"Polyhedral"]

GraphData[{"Antelope",{13,13}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {14 , 14}}

Names

GraphData[{"Antelope",{14 ,14}},"StandardName"]

GraphData[{"Antelope",{14 ,14}},"StandardNames"]

Primary

GraphData[{"Antelope",{14 ,14}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{14 ,14}},"Configuration"]

GraphData[{"Antelope",{14 ,14}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{14 ,14}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{14 ,14}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{14 ,14}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{14 ,14}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{14 ,14}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{14 ,14}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{14 ,14}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{14 ,14}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{14 ,14}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{14 ,14}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{14 ,14}},"Hamiltonian"]

GraphData[{"Antelope",{14 ,14}},"LCF"]

GraphData[{"Antelope",{14 ,14}},"HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData[{"Antelope",{14 ,14}}],All][[All,All,1]]]//Timing

circ=CircularEmbeddingFromHamiltonianCycle[GraphData[{"Antelope",{14 ,14}}],#]&/@hc;

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

gs=bilat

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

quad

Planar

GraphData[{"Antelope",{14 ,14}},"Planar"]

GraphData[{"Antelope",{14 ,14}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{14 ,14}},{"RCN","CN"}]

GraphData[{"Antelope",{14 ,14}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{14 ,14}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{14 ,14}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{14 ,14}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{14 ,14}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{14 ,14}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{14 ,14}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{14 ,14}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{14 ,14}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{14 ,14}},"Circulant"]

GraphData[{"Antelope",{14 ,14}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{14 ,14}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{14 ,14}},"Line"]

LineGraphQ[GraphData[{"Antelope",{14 ,14}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{14 ,14}}&]

RootGraph[GraphData[{"Antelope",{14 ,14}}]]

Halin

GraphData[{"Antelope",{14 ,14}},"Halin"]

GraphData[{"Antelope",{14 ,14}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{14 ,14}},"Rigid"]

GraphData[{"Antelope",{14 ,14}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{14 ,14}},"Flexible"]

GraphData[{"Antelope",{14 ,14}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{14 ,14}},"LCF"]

GraphData[{"Antelope",{14 ,14}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{14 ,14}},"LCFSignature"]

GraphData[{"Antelope",{14 ,14}},"LCFNotations"]

GraphData[{"Antelope",{14 ,14}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{14 ,14}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{14 ,14}}],Debug->True]//Timing

GraphData[{"Antelope",{14 ,14}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{14 ,14}},"VertexList"],GraphData[{"Antelope",{14 ,14}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{14 ,14}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{14 ,14}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{14 ,14}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{14 ,14}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{14 ,14}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{14 ,14}},"Matchstick"]

GraphData[{"Antelope",{14 ,14}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{14 ,14}},"Edges"]]

3D All

GraphData[{"Antelope",{14 ,14}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{14 ,14}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{14 ,14}},"Polyhedral"]

GraphData[{"Antelope",{14 ,14}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {15, 15}}

Names

GraphData[{"Antelope",{15,15}},"StandardName"]

GraphData[{"Antelope",{15,15}},"StandardNames"]

Primary

GraphData[{"Antelope",{15,15}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{15,15}},"Configuration"]

GraphData[{"Antelope",{15,15}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{15,15}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{15,15}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{15,15}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{15,15}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{15,15}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{15,15}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{15,15}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{15,15}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{15,15}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{15,15}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{15,15}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{15,15}},"Planar"]

GraphData[{"Antelope",{15,15}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{15,15}},{"RCN","CN"}]

GraphData[{"Antelope",{15,15}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{15,15}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{15,15}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{15,15}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{15,15}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{15,15}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{15,15}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{15,15}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{15,15}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{15,15}},"Circulant"]

GraphData[{"Antelope",{15,15}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{15,15}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{15,15}},"Line"]

LineGraphQ[GraphData[{"Antelope",{15,15}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{15,15}}&]

RootGraph[GraphData[{"Antelope",{15,15}}]]

Halin

GraphData[{"Antelope",{15,15}},"Halin"]

GraphData[{"Antelope",{15,15}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{15,15}},"Rigid"]

GraphData[{"Antelope",{15,15}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{15,15}},"Flexible"]

GraphData[{"Antelope",{15,15}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{15,15}},"LCF"]

GraphData[{"Antelope",{15,15}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{15,15}},"LCFSignature"]

GraphData[{"Antelope",{15,15}},"LCFNotations"]

GraphData[{"Antelope",{15,15}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{15,15}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{15,15}}],Debug->True]//Timing

GraphData[{"Antelope",{15,15}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{15,15}},"VertexList"],GraphData[{"Antelope",{15,15}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{15,15}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{15,15}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{15,15}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{15,15}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{15,15}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{15,15}},"Matchstick"]

GraphData[{"Antelope",{15,15}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{15,15}},"Edges"]]

3D All

GraphData[{"Antelope",{15,15}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{15,15}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{15,15}},"Polyhedral"]

GraphData[{"Antelope",{15,15}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {16, 16}}

Names

GraphData[{"Antelope",{16,16}},"StandardName"]

GraphData[{"Antelope",{16,16}},"StandardNames"]

Primary

GraphData[{"Antelope",{16,16}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{16,16}},"Configuration"]

GraphData[{"Antelope",{16,16}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{16,16}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{16,16}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{16,16}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{16,16}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{16,16}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{16,16}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{16,16}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{16,16}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{16,16}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{16,16}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{16,16}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{16,16}},"Planar"]

GraphData[{"Antelope",{16,16}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{16,16}},{"RCN","CN"}]

GraphData[{"Antelope",{16,16}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{16,16}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{16,16}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{16,16}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{16,16}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{16,16}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{16,16}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{16,16}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{16,16}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{16,16}},"Circulant"]

GraphData[{"Antelope",{16,16}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{16,16}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{16,16}},"Line"]

LineGraphQ[GraphData[{"Antelope",{16,16}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{16,16}}&]

RootGraph[GraphData[{"Antelope",{16,16}}]]

Halin

GraphData[{"Antelope",{16,16}},"Halin"]

GraphData[{"Antelope",{16,16}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{16,16}},"Rigid"]

GraphData[{"Antelope",{16,16}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{16,16}},"Flexible"]

GraphData[{"Antelope",{16,16}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{16,16}},"LCF"]

GraphData[{"Antelope",{16,16}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{16,16}},"LCFSignature"]

GraphData[{"Antelope",{16,16}},"LCFNotations"]

GraphData[{"Antelope",{16,16}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{16,16}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{16,16}}],Debug->True]//Timing

GraphData[{"Antelope",{16,16}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{16,16}},"VertexList"],GraphData[{"Antelope",{16,16}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{16,16}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{16,16}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{16,16}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{16,16}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{16,16}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{16,16}},"Matchstick"]

GraphData[{"Antelope",{16,16}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{16,16}},"Edges"]]

3D All

GraphData[{"Antelope",{16,16}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{16,16}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{16,16}},"Polyhedral"]

GraphData[{"Antelope",{16,16}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {17, 17}}

Names

GraphData[{"Antelope",{17,17}},"StandardName"]

GraphData[{"Antelope",{17,17}},"StandardNames"]

Primary

GraphData[{"Antelope",{17,17}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{17,17}},"Configuration"]

GraphData[{"Antelope",{17,17}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{17,17}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{17,17}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{17,17}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{17,17}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{17,17}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{17,17}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{17,17}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{17,17}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{17,17}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{17,17}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{17,17}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{17,17}},"Planar"]

GraphData[{"Antelope",{17,17}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{17,17}},{"RCN","CN"}]

GraphData[{"Antelope",{17,17}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{17,17}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{17,17}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{17,17}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{17,17}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{17,17}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{17,17}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{17,17}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{17,17}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{17,17}},"Circulant"]

GraphData[{"Antelope",{17,17}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{17,17}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{17,17}},"Line"]

LineGraphQ[GraphData[{"Antelope",{17,17}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{17,17}}&]

RootGraph[GraphData[{"Antelope",{17,17}}]]

Halin

GraphData[{"Antelope",{17,17}},"Halin"]

GraphData[{"Antelope",{17,17}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{17,17}},"Rigid"]

GraphData[{"Antelope",{17,17}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{17,17}},"Flexible"]

GraphData[{"Antelope",{17,17}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{17,17}},"LCF"]

GraphData[{"Antelope",{17,17}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{17,17}},"LCFSignature"]

GraphData[{"Antelope",{17,17}},"LCFNotations"]

GraphData[{"Antelope",{17,17}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{17,17}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{17,17}}],Debug->True]//Timing

GraphData[{"Antelope",{17,17}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{17,17}},"VertexList"],GraphData[{"Antelope",{17,17}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{17,17}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{17,17}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{17,17}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{17,17}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{17,17}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{17,17}},"Matchstick"]

GraphData[{"Antelope",{17,17}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{17,17}},"Edges"]]

3D All

GraphData[{"Antelope",{17,17}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{17,17}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{17,17}},"Polyhedral"]

GraphData[{"Antelope",{17,17}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {18, 18}}

Names

GraphData[{"Antelope",{18,18}},"StandardName"]

GraphData[{"Antelope",{18,18}},"StandardNames"]

Primary

GraphData[{"Antelope",{18,18}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{18,18}},"Configuration"]

GraphData[{"Antelope",{18,18}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{18,18}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{18,18}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{18,18}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{18,18}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{18,18}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{18,18}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{18,18}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{18,18}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{18,18}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{18,18}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{18,18}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{18,18}},"Planar"]

GraphData[{"Antelope",{18,18}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{18,18}},{"RCN","CN"}]

GraphData[{"Antelope",{18,18}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{18,18}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{18,18}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{18,18}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{18,18}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{18,18}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{18,18}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{18,18}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{18,18}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{18,18}},"Circulant"]

GraphData[{"Antelope",{18,18}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{18,18}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{18,18}},"Line"]

LineGraphQ[GraphData[{"Antelope",{18,18}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{18,18}}&]

RootGraph[GraphData[{"Antelope",{18,18}}]]

Halin

GraphData[{"Antelope",{18,18}},"Halin"]

GraphData[{"Antelope",{18,18}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{18,18}},"Rigid"]

GraphData[{"Antelope",{18,18}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{18,18}},"Flexible"]

GraphData[{"Antelope",{18,18}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{18,18}},"LCF"]

GraphData[{"Antelope",{18,18}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{18,18}},"LCFSignature"]

GraphData[{"Antelope",{18,18}},"LCFNotations"]

GraphData[{"Antelope",{18,18}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{18,18}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{18,18}}],Debug->True]//Timing

GraphData[{"Antelope",{18,18}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{18,18}},"VertexList"],GraphData[{"Antelope",{18,18}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{18,18}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{18,18}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{18,18}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{18,18}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{18,18}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{18,18}},"Matchstick"]

GraphData[{"Antelope",{18,18}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{18,18}},"Edges"]]

3D All

GraphData[{"Antelope",{18,18}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{18,18}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{18,18}},"Polyhedral"]

GraphData[{"Antelope",{18,18}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {19, 19}}

Names

GraphData[{"Antelope",{19,19}},"StandardName"]

GraphData[{"Antelope",{19,19}},"StandardNames"]

Primary

GraphData[{"Antelope",{19,19}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{19,19}},"Configuration"]

GraphData[{"Antelope",{19,19}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{19,19}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{19,19}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{19,19}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{19,19}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{19,19}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{19,19}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{19,19}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{19,19}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{19,19}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{19,19}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{19,19}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{19,19}},"Planar"]

GraphData[{"Antelope",{19,19}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{19,19}},{"RCN","CN"}]

GraphData[{"Antelope",{19,19}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{19,19}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{19,19}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{19,19}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{19,19}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{19,19}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{19,19}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{19,19}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{19,19}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{19,19}},"Circulant"]

GraphData[{"Antelope",{19,19}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{19,19}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{19,19}},"Line"]

LineGraphQ[GraphData[{"Antelope",{19,19}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{19,19}}&]

RootGraph[GraphData[{"Antelope",{19,19}}]]

Halin

GraphData[{"Antelope",{19,19}},"Halin"]

GraphData[{"Antelope",{19,19}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{19,19}},"Rigid"]

GraphData[{"Antelope",{19,19}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{19,19}},"Flexible"]

GraphData[{"Antelope",{19,19}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{19,19}},"LCF"]

GraphData[{"Antelope",{19,19}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{19,19}},"LCFSignature"]

GraphData[{"Antelope",{19,19}},"LCFNotations"]

GraphData[{"Antelope",{19,19}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{19,19}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{19,19}}],Debug->True]//Timing

GraphData[{"Antelope",{19,19}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{19,19}},"VertexList"],GraphData[{"Antelope",{19,19}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{19,19}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{19,19}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{19,19}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{19,19}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{19,19}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{19,19}},"Matchstick"]

GraphData[{"Antelope",{19,19}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{19,19}},"Edges"]]

3D All

GraphData[{"Antelope",{19,19}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{19,19}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{19,19}},"Polyhedral"]

GraphData[{"Antelope",{19,19}},"Graph",{"3D","Polyhedron"}]

{"Antelope", {20, 20}}

Names

GraphData[{"Antelope",{20,20}},"StandardName"]

GraphData[{"Antelope",{20,20}},"StandardNames"]

Primary

GraphData[{"Antelope",{20,20}}]//StyleGraphs[#,ImageSize->Small]&

Configuration

GraphData[{"Antelope",{20,20}},"Configuration"]

GraphData[{"Antelope",{20,20}},"Graph","Configuration"]//StyleGraphs

All

GraphData[{"Antelope",{20,20}},"Graph","All"]//StyleGraphs

Grid

GraphData[{"Antelope",{20,20}},"Graph","Grid"]//StyleGraphs

Inexact

Select[GraphData[{"Antelope",{20,20}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"Antelope",{20,20}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"Antelope",{20,20}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"Antelope",{20,20}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"Antelope",{20,20}},"Graph","Degenerate"]

Perspective

GraphData[{"Antelope",{20,20}},"Graph","Perspective"]//StyleGraphs

Bilateral

GraphData[{"Antelope",{20,20}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"Antelope",{20,20}},"Graph","Circular"]//StyleGraphs

Circular construction

GraphData[{"Antelope",{20,20}},"Hamiltonian"]

Planar

GraphData[{"Antelope",{20,20}},"Planar"]

GraphData[{"Antelope",{20,20}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"Antelope",{20,20}},{"RCN","CN"}]

GraphData[{"Antelope",{20,20}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData[{"Antelope",{20,20}}],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{20,20}}],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{20,20}}],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData[{"Antelope",{20,20}}],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData[{"Antelope",{20,20}}],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Antelope",{20,20}}],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData[{"Antelope",{20,20}},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Antelope",{20,20}}],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"Antelope",{20,20}},"Circulant"]

GraphData[{"Antelope",{20,20}},"Graph","Circulant"]//StyleGraphs

Complement graph

GraphData[{"Antelope",{20,20}},"ComplementGraph","Name"]

Line graph

GraphData[{"Antelope",{20,20}},"Line"]

LineGraphQ[GraphData[{"Antelope",{20,20}}]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== {"Antelope",{20,20}}&]

RootGraph[GraphData[{"Antelope",{20,20}}]]

Halin

GraphData[{"Antelope",{20,20}},"Halin"]

GraphData[{"Antelope",{20,20}},"Graph","Halin"]//StyleGraphs

Rigid

GraphData[{"Antelope",{20,20}},"Rigid"]

GraphData[{"Antelope",{20,20}},"Graph","Rigid"]//StyleGraphs

Flexible

GraphData[{"Antelope",{20,20}},"Flexible"]

GraphData[{"Antelope",{20,20}},"Graph","Flexible"]//StyleGraphs

LCF

GraphData[{"Antelope",{20,20}},"LCF"]

GraphData[{"Antelope",{20,20}},"HamiltonianCycleCount"]

GraphData[{"Antelope",{20,20}},"LCFSignature"]

GraphData[{"Antelope",{20,20}},"LCFNotations"]

GraphData[{"Antelope",{20,20}},"Graph","LCF"]//StyleGraphs

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

UnitDistance

GraphData[{"Antelope",{20,20}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"Antelope",{20,20}}],Debug->True]//Timing

GraphData[{"Antelope",{20,20}},"Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData[{"Antelope",{20,20}},"VertexList"],GraphData[{"Antelope",{20,20}},"EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData[{"Antelope",{20,20}},"Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData[{"Antelope",{20,20}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData[{"Antelope",{20,20}}],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Antelope",{20,20}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Antelope",{20,20}}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

GraphData[{"Antelope",{20,20}},"Matchstick"]

GraphData[{"Antelope",{20,20}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"Antelope",{20,20}},"Edges"]]

3D All

GraphData[{"Antelope",{20,20}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"Antelope",{20,20}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"Antelope",{20,20}},"Polyhedral"]

GraphData[{"Antelope",{20,20}},"Graph",{"3D","Polyhedron"}]

Properties m×n

Connected

GraphData[{"Antelope","Connected"}]

Regular

GraphData[{"Antelope","Regular"}]

Properties n×n

Acyclic

seq=Table[GraphData[{"Antelope",{n,n}},"Acyclic"],{n,15}]

AdjacencyMatrixCount

seq=Table[GraphData[{"Antelope",{n,n}},"AdjacencyMatrixCount"],{n,15}]

Table[(n^2)!/Piecewise[MapIndexed[{#,n==#2[[1]]}&,{1,24,362880,20922789888000,2391175987200,3538944,24576,64,2048,2097152,2048,32}],8],{n,15}]

ArcTransitive

seq=Table[GraphData[{"Antelope",{n,n}},"ArcTransitive"],{n,15}]

ArcTransitivity

seq=Table[GraphData[{"Antelope",{n,n}},"ArcTransitivity"],{n,15}]

AutomorphismCount

seq=Table[GraphData[{"Antelope",{n,n}},"AutomorphismCount"],{n,15}]

Table[GroupOrder[GraphAutomorphismGroup[AntelopeGraph[n,n]]],{n,15}]//Timing

(seq=Table[GroupOrder[GraphAutomorphismGroup[AntelopeGraph[n,n]]],{n,25}])//Timing

Position[seq,8]//Flatten

FindSequenceFunction[Take[seq,12],n]

Piecewise[MapIndexed[{#,n==#2[[1]]}&,{1,24,362880,20922789888000,2391175987200,3538944,24576,64,2048,2097152,2048,32}],8]

gf=x FindGeneratingFunction[Take[seq,12],x]

BalabanIndex

seq=Table[GraphData[{"Antelope",{n,n}},"BalabanIndex"],{n,10}]

Bandwidth

seq=Table[GraphData[{"Antelope",{n,n}},"Bandwidth"],{n,15}]

BridgeCount

seq=Table[GraphData[{"Antelope",{n,n}},"BridgeCount"],{n,15}]

BurningNumber

seq=Table[GraphData[{"Antelope",{n,n}},"BurningNumber"],{n,15}]

CharacteristicPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"CharacteristicPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Chordal

seq=Table[GraphData[{"Antelope",{n,n}},"Chordal"],{n,15}]

ChordCount

seq=Table[GraphData[{"Antelope",{n,n}},"ChordCount"],{n,15}]

Table[Piecewise[{{0,n≤7},{16,n==8}},4 (-6-7 n+(n^2))],{n,15}]

Chordless

seq=Table[GraphData[{"Antelope",{n,n}},"Chordless"],{n,15}]

ChordlessCycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"ChordlessCycleCount"],{n,15}]

FindSequenceFunction[seq,n]

Chords

seq=Table[GraphData[{"Antelope",{n,n}},"Chords"],{n,15}]

ChromaticallyUnique

seq=Table[GraphData[{"Antelope",{n,n}},"ChromaticallyUnique"],{n,15}]

GraphData[{"ChromaticallyUnique","Antelope"}]

ChromaticInvariant

seq=Table[GraphData[{"Antelope",{n,n}},"ChromaticInvariant"],{n,15}]

FindSequenceFunction[Drop[seq,2],n]

ChromaticNumber

seq=Table[GraphData[{"Antelope",{n,n}},"ChromaticNumber"],{n,15}]

Table[ChromaticNumber[AntelopeGraph[n,n]],{n,15}]//Timing

Table[ChromaticNumber[AntelopeGraph[n,n]],{n,20}]//Timing

Table[ChromaticNumber[AntelopeGraph[n,n]],{n,30}]//Timing

ChromaticPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"ChromaticPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Circumference

A301575

seq=Table[GraphData[{"Antelope",{n,n}},"Circumference"],{n,15}]

Grid[MapIndexed[{#2[[1]],#}&,seq],Dividers->All,Alignment->Decimal]

Position[Table[GraphData[{"Antelope",{n,n}},"Connected"],{n,15}],True]//Flatten

Position[Table[GraphData[{"Antelope",{n,n}},"Hamiltonian"],{n,15}],True]//Flatten

Classes

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

CliqueCount

seq=Table[GraphData[{"Antelope",{n,n}},"CliqueCount"],{n,15}]

Table[Length@FindAllCliques[AntelopeGraph[n,n]],{n,15}]//Timing

(seq=Table[Length@FindAllCliques[AntelopeGraph[n,n]],{n,20}])//Timing

(seq=Table[Length@FindAllCliques[AntelopeGraph[n,n]],{n,25}])//Timing

FindSequenceFunction[Drop[seq,2],n-2]//Factor

CliqueCoveringNumber

seq=Table[GraphData[{"Antelope",{n,n}},"CliqueCoveringNumber"],{n,15}]

Table[CliqueCoveringNumber[AntelopeGraph[n,n]],{n,10}]//Timing

FindSequenceFunction[seq,n]

CliqueNumber

seq=Table[GraphData[{"Antelope",{n,n}},"CliqueNumber"],{n,15}]

Table[CliqueNumber[AntelopeGraph[n,n]],{n,20}]//Timing

Table[CliqueNumber[AntelopeGraph[n,n]],{n,30}]//Timing

CliquePolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"CliquePolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

CochromaticGraphNames

Table[GraphData[{"Antelope",{n,n}},"CochromaticGraphNames"],{n,15}]

Select[Select[GraphData["Antelope"],Cases[GraphData[#,"StandardNames"],{"Antelope",{n_,n_}}]=!={}&],GraphData[#,"ChromaticallyUnique"]=!=False&]

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"ComplementOddChordlessCycleCount"],{n,15}]

ConnectedComponentCount

seq=Table[GraphData[{"Antelope",{n,n}},"ConnectedComponentCount"],{n,15}]

Table[ConnectedComponentCount[AntelopeGraph[n,n]],{n,20}]

Position[seq,1]//Flatten

ConnectedDominatingSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"ConnectedDominatingSetCount"],{n,15}]

FindSequenceFunction[seq,n]

ConnectedDominationNumber

seq=Table[GraphData[{"Antelope",{n,n}},"ConnectedDominationNumber"],{n,15}]

FindSequenceFunction[seq,n]

ConnectedDominationPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"ConnectedDominationPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"Antelope",{n,n}},"ConnectedInducedSubgraphCount"],{n,15}]

FindSequenceFunction[DeleteMissing[seq],n]

ConnectedInducedSubgraphPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"CharacteristicPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,6}]

FullSimplify[poly2-poly]

CrossingNumber

seq=Table[GraphData[{"Antelope",{n,n}},"CrossingNumber"],{n,15}]

CycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"CycleCount"],{n,15}]

(x^8) (1-x+(x^2)) (1+x+(x^2)) (1-(x^2)+(x^4)) (1-(x^4)+(x^8))//Expand

CyclePolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"CyclePolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Cyclic

Table[GraphData[{"Antelope",{n,n}},"Cyclic"],{n,15}]

CyclomaticNumber

seq=Table[GraphData[{"Antelope",{n,n}},"CyclomaticNumber"],{n,15}]

(seq=Table[CyclomaticNumber[AntelopeGraph[n,n]],{n,20}])//Timing

FindSequenceFunction[Drop[seq,7],n-7]//Factor

DetourIndex

seq=Table[GraphData[{"Antelope",{n,n}},"DetourIndex"],{n,15}]

DetourPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"DetourPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Diameter

seq=Table[GraphData[{"Antelope",{n,n}},"Diameter"],{n,15}]

Table[GraphDiameter[AntelopeGraph[n,n]],{n,25}]//Timing

Table[GraphDiameter[AntelopeGraph[n,n]],{n,30}]//Timing

Table[GraphDiameter[AntelopeGraph[n,n]],{n,40}]//Timing

DistanceMatrix

MatrixForm/@Table[GraphData[{"Antelope",{n,n}},"DistanceMatrix"],{n,15}]

MatrixForm/@Table[SparseArray[{i_,j_}:>Abs[i-j],n],{n,15}]

DistancePolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"DistancePolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

DistinguishingNumber

seq=Table[GraphData[{"Antelope",{n,n}},"DistinguishingNumber"],{n,15}]

DominatingSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"DominatingSetCount"],{n,15}]

DominationNumber

seq=Table[GraphData[{"Antelope",{n,n}},"DominationNumber"],{n,15}]

Table[DominationNumber[AntelopeGraph[n,n]],{n,15}]//Timing

Table[DominationNumber[AntelopeGraph[n,n]],{n,20}]//Timing

DominationPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"DominationPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

EdgeChromaticNumber

seq=Table[GraphData[{"Antelope",{n,n}},"EdgeChromaticNumber"],{n,15}]

Table[EdgeChromaticNumber[AntelopeGraph[n,n]],{n,10}]//Timing

EdgeConnectivity

Table[GraphData[{"Antelope",{n,n}},"EdgeConnectivity"],{n,15}]

Table[System`EdgeConnectivity[AntelopeGraph[n,n]],{n,2,20}]//Timing

Table[System`EdgeConnectivity[AntelopeGraph[n,n]],{n,2,30}]//Timing

EdgeCount

seq=Table[GraphData[{"Antelope",{n,n}},"EdgeCount"],{n,15}]

(seq=Table[EdgeCount[AntelopeGraph[n,n]],{n,20}])//Timing

FindSequenceFunction[Drop[seq,2],n-2]//Factor

EdgeCoverCount

seq=Table[GraphData[{"Antelope",{n,n}},"EdgeCoverCount"],{n,15}]

EdgeCoverNumber

seq=Table[GraphData[{"Antelope",{n,n}},"EdgeCoverNumber"],{n,15}]

(seq=Table[EdgeCoverNumber[AntelopeGraph[n,n]],{n,30}])//Timing

FindSequenceFunction[Drop[seq,7],n-7]

EdgeCoverPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"EdgeCoverPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

EulerianCycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"EulerianCycleCount"],{n,15}]

FaceCount

seq=Table[GraphData[{"Antelope",{n,n}},"FaceCount"],{n,15}]

FlowPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"FlowPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

FractionalChromaticNumber

seq=Table[GraphData[{"Antelope",{n,n}},"FractionalChromaticNumber"],{n,15}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Antelope",{n,n}},"FractionalEdgeChromaticNumber"],{n,15}]

Girth

seq=Table[GraphData[{"Antelope",{n,n}},"Girth"],{n,15}]

Gonality

seq=Table[GraphData[{"Antelope",{n,n}},"Gonality"],{n,15}]

HamiltonConnected

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonConnected"],{n,15}]

HamiltonDecompositionCount

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonDecompositionCount"],{n,15}]

Hamiltonian

Position[Table[GraphData[{"Antelope",{n,n}},"Hamiltonian"],{n,15}],True]//Flatten

Position[seq,True]//Flatten

Select[GraphData["Antelope"],Head[GraphData[#,"Hamiltonian"]]===Missing&]

Do[Print[{n,TimeConstrained[HamiltonianGraphQ[AntelopeGraph[n,n]],3600]//Timing}],{n,25}];

Do[Print[{n,RandomizedHamiltonianGraphQ[AntelopeGraph[n,n],MaxIterations->1000]//Timing}],{n,25}];

HamiltonianCycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonianCycleCount"],{n,15}]

HamiltonianNumber

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonianNumber"],{n,15}]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

HamiltonianPathCount

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonianPathCount"],{n,15}]

HamiltonianWalkCount

seq=Table[GraphData[{"Antelope",{n,n}},"HamiltonianWalkCount"],{n,15}]

HararyIndex

seq=Table[GraphData[{"Antelope",{n,n}},"HararyIndex"],{n,15}]

seq=Table[HararyIndex[AntelopeGraph[n,n]],{n,20}]

FindSequenceFunction[Drop[seq,7],n-7]

FindSequenceFunction[Drop[seq,8],n-8]

HexagonCount

seq=Table[GraphData[{"Antelope",{n,n}},"HexagonCount"],{n,15}]

seq=Table[CycleCount[AntelopeGraph[n,n],6],{n,20}]

FindSequenceFunction[Drop[seq,8],n-8]

gf=(x^8) FindGeneratingFunction[Drop[seq,8],x]

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

IdiosyncraticPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"IdiosyncraticPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x,y]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceNumber

A286069

seq=Table[GraphData[{"Antelope",{n,n}},"IndependenceNumber"],{n,15}]

FindSequenceFunction[seq,n]

Table[IndependenceNumber[AntelopeGraph[n,n]],{n,10}]//Timing

Table[IndependenceNumber[AntelopeGraph[n,n]],{n,12}]//Timing

Table[IndependenceNumber[AntelopeGraph[n,n]],{n,14}]//Timing

IndependencePolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"IndependencePolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"Antelope",{n,n}},"IndependenceRatio"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]

IndependentEdgeSetCount

A285997

seq=Table[GraphData[{"Antelope",{n,n}},"IndependentEdgeSetCount"],{n,10}]

FindLinearRecurrence[DeleteMissing[Drop[seq,4]]]

FindSequenceFunction[DeleteMissing[seq],n]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

IndependentVertexSetCount

A286039

seq=Table[GraphData[{"Antelope",{n,n}},"IndependentVertexSetCount"],{n,10}]

FindLinearRecurrence[DeleteMissing[seq]]

FindSequenceFunction[DeleteMissing[seq],n]

gf=x FindGeneratingFunction[DeleteMissing[seq],x]

IntersectionNumber

seq=Table[GraphData[{"Antelope",{n,n}},"IntersectionNumber"],{n,15}]

KirchhoffIndex

seq=Table[GraphData[{"Antelope",{n,n}},"KirchhoffIndex"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]

KirchhoffSumIndex

seq=Table[GraphData[{"Antelope",{n,n}},"KirchhoffSumIndex"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]

LaplacianMatrix

MatrixForm/@Table[GraphData[{"Antelope",{n,n}},"LaplacianMatrix"],{n,15}]

LaplacianPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"LaplacianPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[Drop[poly,7]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

LeafCount

seq=Table[GraphData[{"Antelope",{n,n}},"LeafCount"],{n,15}]

LongestCycleCount

A301576

seq=Table[GraphData[{"Antelope",{n,n}},"LongestCycleCount"],{n,15}]

LovaszNumber

seq=Table[GraphData[{"Antelope",{n,n}},"LovaszNumber"],{n,15}]

MatchingGeneratingPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"MatchingGeneratingPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@Drop[poly,4]]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

MatchingNumber

A286040

seq=Table[GraphData[{"Antelope",{n,n}},"MatchingNumber"],{n,15}]

Table[MatchingNumber[AntelopeGraph[n,n]],{n,20}]//Timing

Table[MatchingNumber[AntelopeGraph[n,n]],{n,30}]//Timing

Table[MatchingNumber[AntelopeGraph[n,n]],{n,40}]//Timing

FindSequenceFunction[seq,n]

gf=(x^5) FindGeneratingFunction[Drop[seq,4],x]

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

MatchingPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"MatchingPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

MaximalCliqueCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximalCliqueCount"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]//Factor

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximalIndependentEdgeSetCount"],{n,15}]

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximalIndependentVertexSetCount"],{n,15}]

MaximumCliqueCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximumCliqueCount"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]//Factor

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximumIndependentEdgeSetCount"],{n,15}]

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MaximumIndependentVertexSetCount"],{n,15}]

MaximumLeafNumber

seq=Table[GraphData[{"Antelope",{n,n}},"MaximumLeafNumber"],{n,15}]

MaximumVertexDegree

seq=Table[GraphData[{"Antelope",{n,n}},"MaximumVertexDegree"],{n,15}]

Table[Max[VertexDegree[AntelopeGraph[n,n]]],{n,30}]

MeanDistance

seq=Table[GraphData[{"Antelope",{n,n}},"MeanDistance"],{n,15}]

seq=Table[Total[GraphDistanceMatrix[AntelopeGraph[n,n]],2]/(n^2),{n,20}]

FindSequenceFunction[Drop[seq,7],n-7]

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumCliqueCoveringCount"],{n,15}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumConnectedDominatingSetCount"],{n,15}]

FindSequenceFunction[seq,n]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumDistinguishingLabelingCount"],{n,15}]

MinimumDominatingSetCount

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumDominatingSetCount"],{n,15}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumEdgeCoverCount"],{n,15}]

MinimumVertexDegree

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumVertexDegree"],{n,15}]

Table[Min[VertexDegree[AntelopeGraph[n,n]]],{n,30}]

MolecularTopologicalIndex

seq=Table[GraphData[{"Antelope",{n,n}},"MolecularTopologicalIndex"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]

Nonhamiltonian

Table[GraphData[{"Antelope",{n,n}},"Nonhamiltonian"],{n,15}]

NormalizedLaplacianMatrix

MatrixForm/@Table[GraphData[{"Antelope",{n,n}},"NormalizedLaplacianMatrix"],{n,15}]

OddChordlessCycleCount

seq=Table[GraphData[{"Antelope",{n,n}},"OddChordlessCycleCount"],{n,15}]

PathCount

seq=Table[GraphData[{"Antelope",{n,n}},"PathCount"],{n,15}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n]

gf=x FindGeneratingFunction[seq,x]

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

PathPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"PathPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

PentagonCount

seq=Table[GraphData[{"Antelope",{n,n}},"PentagonCount"],{n,15}]

Perfect

Table[GraphData[{"Antelope",{n,n}},"Perfect"],{n,15}]

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Antelope",{n,n}},"ProjectivePlaneCrossingNumber"],{n,15}]

Radius

seq=Table[GraphData[{"Antelope",{n,n}},"Radius"],{n,15}]

Table[GraphRadius[AntelopeGraph[n,n]],{n,20}]

RankPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"RankPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x,y]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"Antelope",{n,n}},"RectilinearCrossingNumber"],{n,15}]

ReliabilityPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"ReliabilityPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

SigmaPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"SigmaPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

SpanningTreeCount

seq=Table[GraphData[{"Antelope",{n,n}},"SpanningTreeCount"],{n,15}]

FindSequenceFunction[Take[seq,7],n]

FindGeneratingFunction[Take[seq,7],x]

SquareCount

seq=Table[GraphData[{"Antelope",{n,n}},"SquareCount"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]//Factor

StabilityIndex

seq=Table[GraphData[{"Antelope",{n,n}},"StabilityIndex"],{n,15}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

TopologicalIndex

seq=Table[GraphData[{"Antelope",{n,n}},"TopologicalIndex"],{n,15}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

ToroidalCrossingNumber

seq=Table[GraphData[{"Antelope",{n,n}},"ToroidalCrossingNumber"],{n,15}]

Traceable

Position[Table[GraphData[{"Antelope",{n,n}},"Traceable"],{n,15}],True]//Flatten

Do[Print[{n,TimeConstrained[RandomizedTraceableGraphQ[AntelopeGraph[n,n],MaxIterations->1000],60]//Timing}],{n,25}];

TriangleCount

seq=Table[GraphData[{"Antelope",{n,n}},"TriangleCount"],{n,15}]

TuttePolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"TuttePolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x,y]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

Untraceable

Table[GraphData[{"Antelope",{n,n}},"Untraceable"],{n,15}]

VertexConnectivity

Table[GraphData[{"Antelope",{n,n}},"VertexConnectivity"],{n,15}]

VertexCount

seq=Table[GraphData[{"Antelope",{n,n}},"VertexCount"],{n,15}]

FindSequenceFunction[seq,n]

VertexCoverNumber

vertex cover number == vertex count - independence number

seq=Table[GraphData[{"Antelope",{n,n}},"VertexCoverNumber"],{n,15}]

FindSequenceFunction[Drop[seq,7],n-7]

VertexCoverPolynomial

(poly=Factor[Table[GraphData[{"Antelope",{n,n}},"VertexCoverPolynomial"],{n,k0=1,kmax=15}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

ToPolynomialRecurrence[p,x,n,coef]

FormattedPolynomialRecurrence[p,x,n,coef]//TraditionalForm

TeXForm[%]

LinearRecurrence[coef,poly,{2-k0,Length[coef]-k0+1}]

FullSimplify[RSolve[Prepend[Table[p[k+k0-1,x]==poly[[k]],{k,Length[coef]}],ToPolynomialRecurrence[p,x,n,coef]],p[n,x],n],n∈Integers&&n>0]

gf=Factor[(z^k0)FindGeneratingFunction[poly,z]]

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

poly2=Table[XXX,{n,k0,kmax}]

FullSimplify[poly2-poly]

VertexDegrees

seq=Table[GraphData[{"Antelope",{n,n}},"VertexDegrees"],{n,15}]

VertexTransitive

Table[GraphData[{"Antelope",{n,n}},"VertexTransitive"],{n,15}]

WienerIndex

A292039

seq=Table[GraphData[{"Antelope",{n,n}},"WienerIndex"],{n,10}]

seq={0,∞,∞,∞,∞,∞,∞,11548,16660,24320,35548,50572,70036,94812,125968,164564,211632,268576,336952,418136,513764,625848,756484,907584,1081368,1280708,1508184,1766420,2058648,2388508,2759076,3174148,3637872,4154736,4728784,5364824,6068036,6844000,7697708,8635076,9662444,10786620,12013632,13350628,14805240,16385612,18098888,19953568,21958704,24123888};

FindLinearRecurrence[Drop[seq,7]]

FindSequenceFunction[Drop[seq,7],n-6]

gf=(x^8)FindGeneratingFunction[Drop[seq,7],x]

WienerSumIndex

seq=Table[GraphData[{"Antelope",{n,n}},"WienerSumIndex"],{n,15}]

seq=Table[WienerSumIndex[AntelopeGraph[n,n]],{n,20}]

FindSequenceFunction[seq,n]

GraphData

GraphDataClassString

GraphDataClassString["Antelope"]

GraphDataString

Rectangular

Do[Print[GraphDataString[AntelopeGraph[m,n],
"Name"->"("<>ToString[m]<>","<>ToString[n]<>")\[Hyphen]antelope tour graph",
"StandardName"->{"Antelope",{m,n}},
"Skip"->{"Perfect"},
"Classes"->{"Antelope"},
"NotationRules"->{"Antelope"->{m,n}},
TimeConstraint->60
]],{n,2,10},{m,2,n}]

Square

GraphData["Antelope"]

GraphData[{"Antelope",{11,11}},"Name"]

<<MathWorld`Graphs`

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

Do[WriteString[file,GraphDataString[
AntelopeGraph[n,n],
"Name"->"("<>ToString[n]<>","<>ToString[n]<>")\[Hyphen]antelope graph",
"StandardName"->{"Antelope",{n,n}},
"Classes"->{"Antelope"},
"Information"->"AntelopeGraph",
"NotationRules"->{"Antelope"->{n,n}},
"Skip"->{"ABCEnergy","Arboricity","ArithmeticGeometricEnergy","ArithmeticGeometricSpectralRadius","BetweennessCentralities","CharacteristicPolynomial","Circumference","CliqueCoveringNumber","ConnectedDominatingSets","ConstructionNumber","ConnectedInducedSubgraphPolynomial","ConnectedIndicedSubgraphs","Curvatures","CyclicEdgeConnectivity","DetourMatrix","EigenvectorCentralities","Energy","Genus","GracefulLabeling","HadwigerNumber","HamiltonianPaths","HamiltonianWalk","HamiltonianWalks","HITSCentralities","IdiosyncraticPolynomial","IndependencePolynomial","LinklesslyEmbeddable","MatchingGeneratingPolynomial","MaximalIndependentEdgeSets","MaximalIndependentVertexSet","MaximalIndependentVertexSets","MaximumIndependentEdgeSetCount","MaximumIndependentEdgeSets","MaximumIndependentVertexSet","MaximumIndependentVertexSetCount","MaximumIndependentVertexSets","MaximumIrredundantSets","MetricDimension","Meyniel","MinimumCyclicEdgeCut","MinimumCyclicEdgeCuts","MinimumEdgeCuts","MinimumWeightFractionalEdgeColoring","MinimumWeightFractionalColoring","Minors","Outerplanar","PageRankCentralities","PathCoveringNumber","PathPolynomial","Paths","ProjectivePlanar","QuadraticEmbeddingConstant","RandicEnergy","RandicSpectralRadius","Skewness","SomborEnergy","SomborSpectralRadius","TotalDominationPolynomial","TuttePolynomial"},
TimeConstraint->10
]],{n,17,20}]

Close[file]