Flower Graph

Author

Eric W. Weisstein
June 25, 2023

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

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

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

Figure

GraphicsRow[Table[GraphData[{"Flower",n}],{n,5,9}]//StyleGraphs,Dividers->All]

GraphicsRow[Table[FlowerGraph[n],{n,5,9}]//StyleGraphs,Dividers->All]

Snark status

GraphData["Flower"]

Select[GraphData["Flower"],!GraphData[#,"Snark"]&]

Alternate definitions

Herbster and Pontil

Gallian

Seoud and Youssef

Seoud and Youssef.  "Harmonious labelling of helms and related graphs".  Unpublished manuscript.  January 2017.

http://dx.doi.org/10.13140/RG.2.2.11041.61282

helmflowers=Table[HelmFlowerGraph[n],{n,3,7}]

EdgeCount/@helmflowers

RecognizeGraph/@helmflowers

Embeddings

<<MathWorld`Graphs`

{"Flower", 5}

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

Primary

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

Graphs.m

g=FlowerGraph[5]//StyleGraphs[#,ImageSize->Small]&

Construction from Read and Wilson [incorrect embedding]

Mean[v[[{4,7}]]]//FullSimplify

v=Join[CirclePoints[10],CirclePoints[(Sqrt[(5/8)-((Sqrt[5])/8)]),5],CirclePoints[1/3,5]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[{1,10},1,15,8],e[10,11,3],e[9,14,6],e[5,12,2],e[7,13,4],e[{16,20},16],{13,18},{14,19},{12,17},{15,20},{11,16}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

SnarkQ[g]

All

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

Inexact

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

Embedding types

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

Default embedding

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

v=Join[CirclePoints[15],CirclePoints[1/4,5]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,15},1,6],e[{16,20},16,2],{7,12},{5,17},{8,18},{11,19},{14,20},{3,13},{10,15},{4,9}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Flower",5}],All][[All,All,1]]]

Planar

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

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

LCF

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot@MinimalCrossingGraph[GraphData[{"Flower",5}],"CrossingNumber"->4,VertexLabels->None]

HighlightGraph[RotateGraph[GraphicsToGraph[,VertexLabels->Automatic],{8,5}],{21,22,24,23}]

v={{0,0},{3/5,0},{1,0},{1,1},{3/5,1},{0,1},{1/5,1/8},{1/5,7/8},Sequence@@Table[{2/5,i/8},{i,7}],{3/5,2/8},{3/5,6/8},{4/5,1/8},{4/5,4/8},{4/5,7/8}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,1,7,9,18,19,20,15,8,11],e[9,10,11,12,13,14,15],{6,8},{7,13},e[2,16,17,5],{4,20},{3,18},{10,16},{14,17},{12,19}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphData[{"Flower",5},"Graph","LCF"]

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

{{"Flower",5},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"Flower",5}],TimeConstraint->300,"Results"->All]//Timing}

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

Matchstick

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

{"Flower", 6}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Circular construction

Length[hc=FindHamiltonianCycle[GraphData[{"Flower",6}],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData[{"Flower",6}],#]&/@hc],CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Flower",6}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Flower",6},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Flower",6}],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union

GraphPlot@

GraphPlot@

v={#/8,#2/5}&@@@{{0,0},{4,0},{8,0},{4,1},{1,1},{7,1},{1,2},{2,2},{3,2},{5,2},{6,2},{7,2},{1,3},{2,3},{3,3},{5,3},{6,3},{7,3},{1,4},{4,4},{7,4},{0,5},{4,5},{8,5}};
g=Graph[Range@Length@v,UndirectedEdge@@@{e[1,2,3,24,23,22,1,5,7,13,19,22],e[24,21,18,12,6,3],e[19,16,17,18],e[21,15,14,13],e[5,10,11,12],e[6,9,8,7],{2,4},{23,20},e[14,20,17],e[8,4,11],{9,15},{10,16}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

RectilinearCrossingCount@g

GraphPlot@RotateGraph[,-Pi/2]

v={{0,0},{7,0},{1,1},{6,1},{2,2},{3,2},{4,2},{5,2},{3,3},{4,3},{1,4},{2,4},{5,4},{6,4},{1,5},{3,5},{4,5},{6,5},{2,6},{3,6},{4,6},{5,6},{0,7},{7,7}}/7;
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,6,7,2,24,23,1,15,16],e[17,18,2],e[15,11,3,4,14,18],e[12,19,20,21,22,13],{23,19},{22,24},e[{11,14}],e[20,16,9],e[10,17,21],e[3,5,9,10,8,4],{5,6},{7,8}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

gs=UnlabelGraph/@{,}

GraphPlot@MinimalCrossingGraph[GraphData[{"Flower",6}],"CrossingNumber"->6,VertexLabels->None]

v=Join[{{0,0},{5/8,0},{1,0},{1,1},{5/8,1},{0,1},{1/8,1/8},{1/8,2/8},{1/8,7/8},{2/8,5/8},{2/8,6/8}},
{3/8,#/8}&/@{2,3,4,5,7},{4/8,#/8}&/@{1,3},{5/8,#/8}&/@{3,5},{6/8,#/8}&/@{2},{7/8,#/8}&/@{1,6,7}
];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,4,5,6,1,9,16,24,23,22,17,7,8,9,6],e[7,1,8,12,21,23,4,24],{17,18},e[2,19,20,5],e[12,13,14,15,16],e[3,22,4],e[8,10,9],e[10,11,23],{15,20},{21,22},e[13,18,19],{11,14}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,RotateGraph[,{152,120}],,,}

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

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

"Flower6WithInteriorHexagram"

Names

GraphData["Flower6WithInteriorHexagram","StandardName"]

GraphData["Flower6WithInteriorHexagram","StandardNames"]

Primary

GraphData["Flower6WithInteriorHexagram"]//StyleGraphs[#,ImageSize->Small]&

All

GraphData["Flower6WithInteriorHexagram","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["Flower6WithInteriorHexagram","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["Flower6WithInteriorHexagram","EdgeList"]]

Default embedding

GraphPlot@GraphData["Flower6WithInteriorHexagram","Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData["Flower6WithInteriorHexagram","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["Flower6WithInteriorHexagram","Graph","Degenerate"]

Bilateral

GraphData["Flower6WithInteriorHexagram","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["Flower6WithInteriorHexagram","Graph","Circular"]//StyleGraphs

Circular construction

Length[hc=FindHamiltonianCycle[GraphData["Flower6WithInteriorHexagram"],All][[All,All,1]]]

Length[bilat=DeleteDuplicatesBy[
DeleteMissing[BilaterallyRotatedGraph@CircularEmbeddingFromHamiltonianCycle[GraphData["Flower6WithInteriorHexagram"],#]&/@hc],
CircularCanonicalGraphEmbedding]]//Timing

gs=bilat

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

quad

Planar

GraphData["Flower6WithInteriorHexagram","Planar"]

GraphData["Flower6WithInteriorHexagram","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["Flower6WithInteriorHexagram",{"RCN","CN"}]

GraphData["Flower6WithInteriorHexagram","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["Flower6WithInteriorHexagram"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["Flower6WithInteriorHexagram"],TimeConstraint->10])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["Flower6WithInteriorHexagram"],"CrossingNumber"->cn])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["Flower6WithInteriorHexagram"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["Flower6WithInteriorHexagram","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["Flower6WithInteriorHexagram"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

GraphPlot@

{#,#2-1}&@@@{{0,1},{4,1},{8,1},{1,2},{7,2},{1,3},{2,3},{5,3},{6,3},{7,3},{2,4},{6,4},{1,5},{2,5},{3,5},{6,5},{7,5}}

v={#/8,#2/6}&@@@{{0,0},{4,0},{8,0},{1,1},{4,1},{7,1},{1,2},{2,2},{5,2},{6,2},{7,2},{2,3},{6,3},{1,4},{2,4},{3,4},{6,4},{7,4},{1,5},{4,5},{7,5},{0,6},{4,6},{8,6}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,3,24,23,22,1,4,9,10,11],e[4,7,14,19,22],e[24,21,18,11,6,3],e[6,12,7,8,12,15,20,23],e[8,5,10],{2,5},e[10,13,17,20],e[14,15,16,12],e[19,13,9],e[16,21],e[17,18,10],{13,18},{7,15}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot@UnlabelGraph@

Circulant

GraphData["Flower6WithInteriorHexagram","Circulant"]

GraphData["Flower6WithInteriorHexagram","Graph","Circulant"]//StyleGraphs

Line graph

GraphData["Flower6WithInteriorHexagram","Line"]

LineGraphQ[GraphData["Flower6WithInteriorHexagram"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "Flower6WithInteriorHexagram"&]

RootGraph[GraphData["Flower6WithInteriorHexagram"]]

Halin

GraphData["Flower6WithInteriorHexagram","Halin"]

GraphData["Flower6WithInteriorHexagram","Graph","Halin"]//StyleGraphs

Rigid

GraphData["Flower6WithInteriorHexagram","Rigid"]

GraphData["Flower6WithInteriorHexagram","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["Flower6WithInteriorHexagram","Flexible"]

GraphData["Flower6WithInteriorHexagram","Graph","Flexible"]//StyleGraphs

LCF

GraphData["Flower6WithInteriorHexagram","LCF"]

GraphData["Flower6WithInteriorHexagram","HamiltonianCycleCount"]

GraphData["Flower6WithInteriorHexagram","LCFSignature"]

GraphData["Flower6WithInteriorHexagram","LCFNotations"]

GraphData["Flower6WithInteriorHexagram","Graph","LCF"]//StyleGraphs

UnitDistance

GraphData["Flower6WithInteriorHexagram","UnitDistance"]

UnitDistanceGraphQ[GraphData["Flower6WithInteriorHexagram"],Debug->True]//Timing

GraphData["Flower6WithInteriorHexagram","Graph","UnitDistance"]//StyleGraphs

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

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["Flower6WithInteriorHexagram","VertexList"],GraphData["Flower6WithInteriorHexagram","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["Flower6WithInteriorHexagram","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["Flower6WithInteriorHexagram"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["Flower6WithInteriorHexagram"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"Flower6WithInteriorHexagram",l=FindSubgraphUnitDistanceEmbedding[GraphData["Flower6WithInteriorHexagram"],TimeConstraint->300,"Results"->All]//Timing}

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

gs={,};

Matchstick

GraphData["Flower6WithInteriorHexagram","Matchstick"]

GraphData["Flower6WithInteriorHexagram","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["Flower6WithInteriorHexagram","Edges"]]

3D All

GraphData["Flower6WithInteriorHexagram","Graph",{"3D","All"}]

3D UnitDistance

GraphData["Flower6WithInteriorHexagram","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["Flower6WithInteriorHexagram","Polyhedral"]

{"Flower", 7}

Names

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

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

Primary

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

Construction from Read and Wilson [incorrect embedding]

Mean@v[[{6,9}]]//FullSimplify

v=Join[CirclePoints[14],CirclePoints[Cos[((3 π)/14)],7],CirclePoints[(1/2),7]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[{1,14},1],e[{22,28},22],e[12,21,1],e[14,15,3],e[2,16,5],e[4,17,7],e[8,19,11],e[9,18,6],e[10,20,13],{20,27},{19,26},{18,25},{17,24},{16,23},{15,22},{21,28}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

SnarkQ[g]

All

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

Inexact

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

Embedding types

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

Default embedding

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

v=Join[CirclePoints[{1,-Pi/2},21],CirclePoints[{1/3,-Pi/2},7]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[18,11,21,1,2,12,5],e[22,25,28,24,27,23,26,22],{1,22},e[5,4,3,17,6,7,8,15,5],{4,23},{25,10},{26,13},{19,28},{16,27},{24,7},e[11,10,9,2],e[12,13,14,21],e[15,16,17],e[18,19,20,9],{3,14},{8,18},{20,6}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

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

GraphPlot@RotateGraph[,4Pi/7]

GraphData[{"Flower", 7},"V"]

v={#/8,#2/7}&@@@{{0,0},{8,0},{2,1},{3,1},{5,1},{6,1},{2,2},{6,2},{0,3},{1,3},{2,3},{6,3},{7,3},{8,3},{2,4},{6,4},{1,6},{2,6},{3,6},{4,6},{5,6},{6,6},{7,6},{0,7},{4,7},{8,7},{2,5},{6,5}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,2,14,26,25,24,9,1,3,4,5,6,2],e[9,10,11],e[12,13,14],e[24,17,18,19,20,21,22,23,26],{20,25},e[17,15,11,7,3],e[23,16,12,8,6],{7,8},{15,21},{19,16},e[18,27,10],e[22,28,13],{4,27},{5,28}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

GraphPlot@RotateGraph[,-Pi/7]

v=({Which[#<3,#,#>7,#+2,True,#+1],#2}&@@@{{0,0},{10,0},{1,1},{3,1},{4,1},{5,1},{6,1},{7,1},{9,1},{3,2},{7,2},{2,4},{5,3},{8,4},{3,4},{7,4},{3,5},{7,5},{1,6},{2,6},{4,6},{6,6},{8,6},{9,6},{0,7},{3,7},{7,7},{10,7}})

v={#/12,#2/7}&@@@{{0,0},{12,0},{1,1},{4,1},{5,1},{6,1},{7,1},{8,1},{11,1},{4,2},{8,2},{3,4},{6,3},{9,4},{4,4},{8,4},{4,5},{8,5},{1,6},{2,6},{5,6},{7,6},{10,6},{11,6},{0,7},{4,7},{8,7},{12,7}};
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[1,7,6,5,2,28,27,26,25,1,3,4,5],e[2,9,8,7],e[25,19,10],e[28,24,11],e[4,10,15,17,26],e[8,11,16,18,27],e[19,20,21,22,23,24],{15,16},{17,21},{18,22},e[3,12,20],e[23,14,9],e[12,13,14],{6,13}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

gs=UnlabelGraph/@{,}

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

g=;

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"Flower", 8}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

GraphPlot@GraphData[{"Flower",8},"Edges"]

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Flower",8}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Flower",8},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Flower",8}],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union

GraphData[{"Flower",8},"V"]

GraphPlot@RotateGraph[,3Pi/8]

RegionIntersection[InfiniteLine@v[[{28,22}]],InfiniteLine@v[[{11,26}]]]//FullSimplify

Mean[v[[{30,11}]]]//FullSimplify

v=Delete[Join[UnitCirclePoints[8],CirclePoints[7/8,24]],List/@{11,12,29,30}];
v=Join[v,{{-((7 (Sqrt[(3/2)]))/(16 (Cos[(π/24)]+Sin[((5 π)/24)]))),-(7/8) Sin[(π/8)]},{((7 (Sqrt[(3/2)]))/(16 (Cos[(π/24)]+Sin[((5 π)/24)]))),-(7/8) Sin[(π/8)]},v[[25]]+{1/5,0},v[[12]]-{1/5,0}}];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,8},1,10],e[8,27,28,9,10],e[2,{11,26},7],{5,20},{4,17},{3,14},{6,23},{21,16},{19,24},{18,13},e[28,29,22],e[9,30,15],e[25,31,32,12],{27,31},{10,32},{26,29},{30,11}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

LCF compute all

Length[FindHamiltonianCycle[GraphData[{"Flower",8}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Flower",8}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

gs={,,,,,};

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"Flower", 9}

Names

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

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

Primary

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

Construction from Read and Wilson [incorrect embedding]

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData[{"Flower",9}],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData[{"Flower",9},"V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData[{"Flower",9}],"CrossingNumber"->cn,RandomSeed->i],{i,500}]//Union

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

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

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"Flower", 10}

Names

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

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

Primary

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

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

Table[MinimalCrossingGraph[GraphData[{"Flower",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[{"Flower",10},"Circulant"]

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

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

LCF compute all

Length[FindHamiltonianCycle[GraphData[{"Flower",10}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"Flower",10}],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

g=;

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"Flower", 11}

Names

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

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

Primary

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

Construction from Read and Wilson [incorrect embedding]

All

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

Inexact

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

Embedding types

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

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Bilateral

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

Circular

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

Planar

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

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

MinimalCrossing

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

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

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

Table[MinimalCrossingGraph[GraphData[{"Flower",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[{"Flower",11},"Circulant"]

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

Line graph

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

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

RootGraph

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

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

Halin

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

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

Rigid

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

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

Flexible

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

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

LCF

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

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

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

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

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

UnitDistance

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

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

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

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

UnitDistance by refinement

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

UnitDistanceGraphEmbeddingQ[%]

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

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

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

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

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

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

Matchstick

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

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

3D default

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

3D All

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

3D UnitDistance

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

3D Polyhedron

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

{"Flower", 12}

GraphData[{"Flower",12},"Graph","MinimalCrossing"]

GraphData[{"Flower",12},"Graph","LCF"]

GraphData[{"Flower",12},"Graph","UnitDistance"]

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

GraphData[{"Flower",12},"Graph","All"]

GraphPlot[GraphData[{"Flower",12},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",12},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",12},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",12}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->2000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}]

{"Flower", 13}

GraphData[{"Flower",13},"Graph","MinimalCrossing"]

GraphData[{"Flower",13},"Graph","LCF"]

GraphData[{"Flower",13},"Graph","UnitDistance"]

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

GraphData[{"Flower",13},"Graph","All"]

GraphPlot[GraphData[{"Flower",13},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",13},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",13},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",13}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->5000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}]

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(RowDefault["-0.885", FEExpressionsNumericalApproximationElider]), Root[-1-6 #1+24 ((#1)^2)+32 ((#1)^3)-80 ((#1)^4)-32 ((#1)^5)+64 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.03", FEExpressionsNumericalApproximationElider]), Root[1-312 ((#1)^2)+11440 ((#1)^4)-109824 ((#1)^6)+329472 ((#1)^8)-292864 ((#1)^10)+53248 ((#1)^12)&,12], ]),(NumericalApproximation[Root, (RowDefault["0.992", FEExpressionsNumericalApproximationElider]), Root[16901-67944 ((#1)^2)+52080 ((#1)^4)+94464 ((#1)^6)-154880 ((#1)^8)+55296 ((#1)^10)+4096 ((#1)^12)&,8], ])},{(NumericalApproximation[Root, (RowDefault["1.28", FEExpressionsNumericalApproximationElider]), Root[5706742876641-3463559611292664 ((#1)^2)+64673206607114352 ((#1)^4)-495836744922509120 ((#1)^6)+2039024590951672064 ((#1)^8)-4977494051475299328 ((#1)^10)+7561995977518374912 ((#1)^12)-7292298907842297856 ((#1)^14)+4463873417954263040 ((#1)^16)-1700202167953522688 ((#1)^18)+384462425964937216 ((#1)^20)-46817701426888704 ((#1)^22)+2353116095709184 ((#1)^24)&,21], ]),(NumericalApproximation[Root, (RowDefault["1.65", FEExpressionsNumericalApproximationElider]), Root[169-5312008 ((#1)^2)+7564296688 ((#1)^4)-1106683135168 ((#1)^6)+26132962948352 ((#1)^8)-158451749753856 ((#1)^10)-187056620969984 ((#1)^12)+1126206549245952 ((#1)^14)+210250291544064 ((#1)^16)-845029680087040 ((#1)^18)-98019267575808 ((#1)^20)+82095124774912 ((#1)^22)+13923763879936 ((#1)^24)&,16], ])},{(NumericalApproximation[Root, (RowDefault["2.15", FEExpressionsNumericalApproximationElider]), Root[4096-13312 ((#1)^2)+16640 ((#1)^4)-9984 ((#1)^6)+2912 ((#1)^8)-364 ((#1)^10)+13 ((#1)^12)&,11], ]),0},{(NumericalApproximation[Root, (RowDefault["1.91", FEExpressionsNumericalApproximationElider]), Root[1-78 ((#1)^2)+715 ((#1)^4)-1716 ((#1)^6)+1287 ((#1)^8)-286 ((#1)^10)+13 ((#1)^12)&,11], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

{"Flower", 14}

GraphData[{"Flower",14},"Graph","MinimalCrossing"]

GraphData[{"Flower",14},"Graph","LCF"]

GraphData[{"Flower",14},"Graph","UnitDistance"]

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

GraphData[{"Flower",14},"Graph","All"]

GraphPlot[GraphData[{"Flower",14},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",14},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",14},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",14}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->5000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}]

v={{(NumericalApproximation[Root, (RowDefault["1.54", FEExpressionsNumericalApproximationElider]), Root[169+1168 ((#1)^2)+29184 ((#1)^4)-107136 ((#1)^6)+111616 ((#1)^8)-39936 ((#1)^10)+4096 ((#1)^12)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.85", FEExpressionsNumericalApproximationElider]), Root[1-28 #1+224 ((#1)^2)-896 ((#1)^3)+1792 ((#1)^4)-1568 ((#1)^5)+448 ((#1)^6)&,4], ])},{(NumericalApproximation[Root, (RowDefault["0.596", FEExpressionsNumericalApproximationElider]), Root[289328879449-30497098677052 ((#1)^2)+629102356723904 ((#1)^4)-4040641450012608 ((#1)^6)+13499939253887744 ((#1)^8)-31115327300908032 ((#1)^10)+51766985564786688 ((#1)^12)-50280584449966080 ((#1)^14)+21934098587254784 ((#1)^16)-4221955471835136 ((#1)^18)+290713366429696 ((#1)^20)+996981866496 ((#1)^22)+40282095616 ((#1)^24)&,11], ]),(NumericalApproximation[Root, (RowDefault["2.17", FEExpressionsNumericalApproximationElider]), Root[-954071-5414010 #1+4747288 ((#1)^2)+33413576 ((#1)^3)-10785712 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(RowDefault["2.02", FEExpressionsNumericalApproximationElider]), Root[-1-8 #1-12 ((#1)^2)+8 ((#1)^3)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-1.46", FEExpressionsNumericalApproximationElider]), Root[1-32 ((#1)^2)+576 ((#1)^4)-4992 ((#1)^6)+16384 ((#1)^8)-15360 ((#1)^10)+4096 ((#1)^12)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.91", FEExpressionsNumericalApproximationElider]), Root[-13+196 #1-616 ((#1)^2)+2016 ((#1)^3)+448 ((#1)^4)-1568 ((#1)^5)+448 ((#1)^6)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-1.98", FEExpressionsNumericalApproximationElider]), Root[199477463641-5984529808208 ((#1)^2)+75269941112800 ((#1)^4)-523369919959936 ((#1)^6)+2227879970109696 ((#1)^8)-6064644430451712 ((#1)^10)+10681691650215936 ((#1)^12)-11948199447969792 ((#1)^14)+7940864716832768 ((#1)^16)-2696882255298560 ((#1)^18)+348329351315456 ((#1)^20)-4753287282688 ((#1)^22)+40282095616 ((#1)^24)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.06", 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((#1)^6)+22528 ((#1)^8)-23552 ((#1)^10)+4096 ((#1)^12)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.09", FEExpressionsNumericalApproximationElider]), Root[-41-308 #1-448 ((#1)^2)+1008 ((#1)^3)+896 ((#1)^4)-1568 ((#1)^5)+448 ((#1)^6)&,2], ])},{(NumericalApproximation[Root, (RowDefault["-2.24", FEExpressionsNumericalApproximationElider]), Root[475109209-13000224522488 ((#1)^2)+118270395847856 ((#1)^4)-163454758564160 ((#1)^6)-587128229619456 ((#1)^8)+106488116568064 ((#1)^10)+6636512967827456 ((#1)^12)-13583800980291584 ((#1)^14)+9877647385690112 ((#1)^16)-2876910493499392 ((#1)^18)+246775670636544 ((#1)^20)+5951679627264 ((#1)^22)+40282095616 ((#1)^24)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.0985", FEExpressionsNumericalApproximationElider]), Root[-345883+7174328 #1-51788212 ((#1)^2)+173657400 ((#1)^3)-268538032 ((#1)^4)+72363200 ((#1)^5)+356720896 ((#1)^6)-523392128 ((#1)^7)+295210496 ((#1)^8)-57526784 ((#1)^9)-6171648 ((#1)^10)+2107392 ((#1)^11)+200704 ((#1)^12)&,2], ])},{(NumericalApproximation[Root, (RowDefault["-1.44", FEExpressionsNumericalApproximationElider]), Root[-1+14 ((#1)^2)-21 ((#1)^4)+7 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.80", FEExpressionsNumericalApproximationElider]), Root[1-2 #1-((#1)^2)+((#1)^3)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-1.29", FEExpressionsNumericalApproximationElider]), Root[-1+56 ((#1)^2)-784 ((#1)^4)+448 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.62", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-2.40", FEExpressionsNumericalApproximationElider]), Root[169+1168 ((#1)^2)+29184 ((#1)^4)-107136 ((#1)^6)+111616 ((#1)^8)-39936 ((#1)^10)+4096 ((#1)^12)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.0554", FEExpressionsNumericalApproximationElider]), Root[1-28 #1+224 ((#1)^2)-896 ((#1)^3)+1792 ((#1)^4)-1568 ((#1)^5)+448 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-2.07", FEExpressionsNumericalApproximationElider]), Root[289328879449-30497098677052 ((#1)^2)+629102356723904 ((#1)^4)-4040641450012608 ((#1)^6)+13499939253887744 ((#1)^8)-31115327300908032 ((#1)^10)+51766985564786688 ((#1)^12)-50280584449966080 ((#1)^14)+21934098587254784 ((#1)^16)-4221955471835136 ((#1)^18)+290713366429696 ((#1)^20)+996981866496 ((#1)^22)+40282095616 ((#1)^24)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.885", FEExpressionsNumericalApproximationElider]), Root[-954071-5414010 #1+4747288 ((#1)^2)+33413576 ((#1)^3)-10785712 ((#1)^4)-78348256 ((#1)^5)+13852608 ((#1)^6)+95804800 ((#1)^7)-27835136 ((#1)^8)-47014912 ((#1)^9)+25840640 ((#1)^10)-3512320 ((#1)^11)+200704 ((#1)^12)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-2.08", FEExpressionsNumericalApproximationElider]), Root[-1+21 ((#1)^2)-35 ((#1)^4)+7 ((#1)^6)&,1], ]),1},{(NumericalApproximation[Root, (RowDefault["-1.87", 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])},{(NumericalApproximation[Root, (RowDefault["-2.30", FEExpressionsNumericalApproximationElider]), Root[-64+112 ((#1)^2)-56 ((#1)^4)+7 ((#1)^6)&,1], ]),0},{(NumericalApproximation[Root, (RowDefault["-2.08", FEExpressionsNumericalApproximationElider]), Root[-1+21 ((#1)^2)-35 ((#1)^4)+7 ((#1)^6)&,1], ]),0},{(NumericalApproximation[Root, (RowDefault["-1.54", FEExpressionsNumericalApproximationElider]), Root[169+1168 ((#1)^2)+29184 ((#1)^4)-107136 ((#1)^6)+111616 ((#1)^8)-39936 ((#1)^10)+4096 ((#1)^12)&,2], ]),(NumericalApproximation[Root, (RowDefault["-1.85", FEExpressionsNumericalApproximationElider]), Root[1+28 #1+224 ((#1)^2)+896 ((#1)^3)+1792 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.596", FEExpressionsNumericalApproximationElider]), Root[289328879449-30497098677052 ((#1)^2)+629102356723904 ((#1)^4)-4040641450012608 ((#1)^6)+13499939253887744 ((#1)^8)-31115327300908032 ((#1)^10)+51766985564786688 ((#1)^12)-50280584449966080 ((#1)^14)+21934098587254784 ((#1)^16)-4221955471835136 ((#1)^18)+290713366429696 ((#1)^20)+996981866496 ((#1)^22)+40282095616 ((#1)^24)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.17", FEExpressionsNumericalApproximationElider]), Root[-954071+5414010 #1+4747288 ((#1)^2)-33413576 ((#1)^3)-10785712 ((#1)^4)+78348256 ((#1)^5)+13852608 ((#1)^6)-95804800 ((#1)^7)-27835136 ((#1)^8)+47014912 ((#1)^9)+25840640 ((#1)^10)+3512320 ((#1)^11)+200704 ((#1)^12)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-2.08", FEExpressionsNumericalApproximationElider]), Root[-1+21 ((#1)^2)-35 ((#1)^4)+7 ((#1)^6)&,1], ]),-1},{(NumericalApproximation[Root, (RowDefault["-1.87", FEExpressionsNumericalApproximationElider]), Root[-1+392 ((#1)^2)-1680 ((#1)^4)+448 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.901", FEExpressionsNumericalApproximationElider]), Root[-1-4 #1+4 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.589", FEExpressionsNumericalApproximationElider]), Root[1-16 ((#1)^2)-640 ((#1)^4)-3200 ((#1)^6)+22528 ((#1)^8)-23552 ((#1)^10)+4096 ((#1)^12)&,3], ]),(NumericalApproximation[Root, (RowDefault["-2.33", FEExpressionsNumericalApproximationElider]), Root[-41+308 #1-448 ((#1)^2)-1008 ((#1)^3)+896 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.403", FEExpressionsNumericalApproximationElider]), Root[475109209-13000224522488 ((#1)^2)+118270395847856 ((#1)^4)-163454758564160 ((#1)^6)-587128229619456 ((#1)^8)+106488116568064 ((#1)^10)+6636512967827456 ((#1)^12)-13583800980291584 ((#1)^14)+9877647385690112 ((#1)^16)-2876910493499392 ((#1)^18)+246775670636544 ((#1)^20)+5951679627264 ((#1)^22)+40282095616 ((#1)^24)&,10], ]),(NumericalApproximation[Root, (RowDefault["-2.21", FEExpressionsNumericalApproximationElider]), Root[-345883-7174328 #1-51788212 ((#1)^2)-173657400 ((#1)^3)-268538032 ((#1)^4)-72363200 ((#1)^5)+356720896 ((#1)^6)+523392128 ((#1)^7)+295210496 ((#1)^8)+57526784 ((#1)^9)-6171648 ((#1)^10)-2107392 ((#1)^11)+200704 ((#1)^12)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.44", FEExpressionsNumericalApproximationElider]), Root[-1+14 ((#1)^2)-21 ((#1)^4)+7 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-1.80", FEExpressionsNumericalApproximationElider]), Root[-1-2 #1+((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.29", FEExpressionsNumericalApproximationElider]), Root[-1+56 ((#1)^2)-784 ((#1)^4)+448 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-1.62", FEExpressionsNumericalApproximationElider]), Root[1+12 #1+20 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.481", FEExpressionsNumericalApproximationElider]), Root[1-32 ((#1)^2)+576 ((#1)^4)-4992 ((#1)^6)+16384 ((#1)^8)-15360 ((#1)^10)+4096 ((#1)^12)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.36", FEExpressionsNumericalApproximationElider]), Root[-13-196 #1-616 ((#1)^2)-2016 ((#1)^3)+448 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.32", FEExpressionsNumericalApproximationElider]), Root[199477463641-5984529808208 ((#1)^2)+75269941112800 ((#1)^4)-523369919959936 ((#1)^6)+2227879970109696 ((#1)^8)-6064644430451712 ((#1)^10)+10681691650215936 ((#1)^12)-11948199447969792 ((#1)^14)+7940864716832768 ((#1)^16)-2696882255298560 ((#1)^18)+348329351315456 ((#1)^20)-4753287282688 ((#1)^22)+40282095616 ((#1)^24)&,15], ]),(NumericalApproximation[Root, (RowDefault["-1.82", FEExpressionsNumericalApproximationElider]), Root[-3138071+21639156 #1-11331432 ((#1)^2)-64628704 ((#1)^3)+82583536 ((#1)^4)+34632416 ((#1)^5)-183107008 ((#1)^6)-34615168 ((#1)^7)+166358528 ((#1)^8)+118741504 ((#1)^9)+29102080 ((#1)^10)+2809856 ((#1)^11)+200704 ((#1)^12)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.513", FEExpressionsNumericalApproximationElider]), Root[-1+7 ((#1)^2)-14 ((#1)^4)+7 ((#1)^6)&,3], ]),(NumericalApproximation[Root, (RowDefault["-2.25", FEExpressionsNumericalApproximationElider]), Root[-1-#1+2 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.462", FEExpressionsNumericalApproximationElider]), Root[-1+56 ((#1)^2)-336 ((#1)^4)+448 ((#1)^6)&,2], ]),(NumericalApproximation[Root, (RowDefault["-2.02", FEExpressionsNumericalApproximationElider]), Root[1-8 #1+12 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.46", FEExpressionsNumericalApproximationElider]), Root[1-32 ((#1)^2)+576 ((#1)^4)-4992 ((#1)^6)+16384 ((#1)^8)-15360 ((#1)^10)+4096 ((#1)^12)&,8], ]),(NumericalApproximation[Root, (RowDefault["-1.91", FEExpressionsNumericalApproximationElider]), Root[-13-196 #1-616 ((#1)^2)-2016 ((#1)^3)+448 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,2], ])},{(NumericalApproximation[Root, (RowDefault["1.98", FEExpressionsNumericalApproximationElider]), Root[199477463641-5984529808208 ((#1)^2)+75269941112800 ((#1)^4)-523369919959936 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((#1)^4)+448 ((#1)^6)&,5], ]),(NumericalApproximation[Root, (RowDefault["-2.02", FEExpressionsNumericalApproximationElider]), Root[1-8 #1+12 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.14", FEExpressionsNumericalApproximationElider]), Root[1-16 ((#1)^2)-640 ((#1)^4)-3200 ((#1)^6)+22528 ((#1)^8)-23552 ((#1)^10)+4096 ((#1)^12)&,8], ]),(NumericalApproximation[Root, (RowDefault["-1.09", FEExpressionsNumericalApproximationElider]), Root[-41+308 #1-448 ((#1)^2)-1008 ((#1)^3)+896 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,3], ])},{(NumericalApproximation[Root, (RowDefault["2.24", FEExpressionsNumericalApproximationElider]), Root[475109209-13000224522488 ((#1)^2)+118270395847856 ((#1)^4)-163454758564160 ((#1)^6)-587128229619456 ((#1)^8)+106488116568064 ((#1)^10)+6636512967827456 ((#1)^12)-13583800980291584 ((#1)^14)+9877647385690112 ((#1)^16)-2876910493499392 ((#1)^18)+246775670636544 ((#1)^20)+5951679627264 ((#1)^22)+40282095616 ((#1)^24)&,16], ]),(NumericalApproximation[Root, (RowDefault["-0.0985", FEExpressionsNumericalApproximationElider]), Root[-345883-7174328 #1-51788212 ((#1)^2)-173657400 ((#1)^3)-268538032 ((#1)^4)-72363200 ((#1)^5)+356720896 ((#1)^6)+523392128 ((#1)^7)+295210496 ((#1)^8)+57526784 ((#1)^9)-6171648 ((#1)^10)-2107392 ((#1)^11)+200704 ((#1)^12)&,7], ])},{(NumericalApproximation[Root, (RowDefault["1.44", FEExpressionsNumericalApproximationElider]), Root[-1+14 ((#1)^2)-21 ((#1)^4)+7 ((#1)^6)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.80", FEExpressionsNumericalApproximationElider]), Root[-1-2 #1+((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.29", FEExpressionsNumericalApproximationElider]), Root[-1+56 ((#1)^2)-784 ((#1)^4)+448 ((#1)^6)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.62", FEExpressionsNumericalApproximationElider]), Root[1+12 #1+20 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.40", FEExpressionsNumericalApproximationElider]), Root[169+1168 ((#1)^2)+29184 ((#1)^4)-107136 ((#1)^6)+111616 ((#1)^8)-39936 ((#1)^10)+4096 ((#1)^12)&,8], ]),(NumericalApproximation[Root, (RowDefault["-0.0554", FEExpressionsNumericalApproximationElider]), Root[1+28 #1+224 ((#1)^2)+896 ((#1)^3)+1792 ((#1)^4)+1568 ((#1)^5)+448 ((#1)^6)&,4], ])},{(NumericalApproximation[Root, (RowDefault["2.07", FEExpressionsNumericalApproximationElider]), Root[289328879449-30497098677052 ((#1)^2)+629102356723904 ((#1)^4)-4040641450012608 ((#1)^6)+13499939253887744 ((#1)^8)-31115327300908032 ((#1)^10)+51766985564786688 ((#1)^12)-50280584449966080 ((#1)^14)+21934098587254784 ((#1)^16)-4221955471835136 ((#1)^18)+290713366429696 ((#1)^20)+996981866496 ((#1)^22)+40282095616 ((#1)^24)&,16], ]),(NumericalApproximation[Root, (RowDefault["0.885", FEExpressionsNumericalApproximationElider]), Root[-954071+5414010 #1+4747288 ((#1)^2)-33413576 ((#1)^3)-10785712 ((#1)^4)+78348256 ((#1)^5)+13852608 ((#1)^6)-95804800 ((#1)^7)-27835136 ((#1)^8)+47014912 ((#1)^9)+25840640 ((#1)^10)+3512320 ((#1)^11)+200704 ((#1)^12)&,8], ])},{(NumericalApproximation[Root, (RowDefault["2.08", FEExpressionsNumericalApproximationElider]), Root[-1+21 ((#1)^2)-35 ((#1)^4)+7 ((#1)^6)&,6], ]),-1},{(NumericalApproximation[Root, (RowDefault["1.87", FEExpressionsNumericalApproximationElider]), Root[-1+392 ((#1)^2)-1680 ((#1)^4)+448 ((#1)^6)&,6], ]),(NumericalApproximation[Root, (RowDefault["-0.901", FEExpressionsNumericalApproximationElider]), Root[-1-4 #1+4 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.19", FEExpressionsNumericalApproximationElider]), Root[-7+140 ((#1)^2)-336 ((#1)^4)+64 ((#1)^6)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.993", FEExpressionsNumericalApproximationElider]), Root[29+308 ((#1)^2)-784 ((#1)^4)+448 ((#1)^6)&,4], ])},{(NumericalApproximation[Root, (RowDefault["1.48", FEExpressionsNumericalApproximationElider]), Root[10004569-117944960 ((#1)^2)+348982816 ((#1)^4)-387911552 ((#1)^6)+156712192 ((#1)^8)-19367936 ((#1)^10)+200704 ((#1)^12)&,10], ]),(NumericalApproximation[Root, (RowDefault["1.69", FEExpressionsNumericalApproximationElider]), Root[1-41104 ((#1)^2)+8809248 ((#1)^4)+6335616 ((#1)^6)-53575424 ((#1)^8)+16959488 ((#1)^10)+200704 ((#1)^12)&,8], ])},{(NumericalApproximation[Root, (RowDefault["2.30", FEExpressionsNumericalApproximationElider]), Root[-64+112 ((#1)^2)-56 ((#1)^4)+7 ((#1)^6)&,6], ]),0},{(NumericalApproximation[Root, (RowDefault["2.08", FEExpressionsNumericalApproximationElider]), Root[-1+21 ((#1)^2)-35 ((#1)^4)+7 ((#1)^6)&,6], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

{"Flower", 15}

GraphData[{"Flower",15},"Graph","MinimalCrossing"]

GraphData[{"Flower",15},"Graph","LCF"]

GraphData[{"Flower",15},"Graph","UnitDistance"]

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

GraphData[{"Flower",15},"Graph","All"]

GraphPlot[GraphData[{"Flower",15},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",15},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",15},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",15}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->2000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}];//Timing

v={{(NumericalApproximation[Root, (RowDefault["1.74", FEExpressionsNumericalApproximationElider]), Root[841-1159480 ((#1)^2)+2050704 ((#1)^4)+2225600 ((#1)^6)-4345344 ((#1)^8)-609280 ((#1)^10)+2207744 ((#1)^12)-737280 ((#1)^14)+65536 ((#1)^16)&,11], ]),(NumericalApproximation[Root, (RowDefault["1.87", FEExpressionsNumericalApproximationElider]), Root[25-800 #1+6940 ((#1)^2)-24120 ((#1)^3)+37216 ((#1)^4)-24928 ((#1)^5)+7104 ((#1)^6)-1152 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["0.825", FEExpressionsNumericalApproximationElider]), Root[10462162449019571161-2159155264986839684636 ((#1)^2)+74160666927567635558016 ((#1)^4)-927758609593085932521472 ((#1)^6)+5842795857580150340071424 ((#1)^8)-21374700168613943938201600 ((#1)^10)+«6»+22355752856510619265794048 ((#1)^20)-6936955686385638105415680 ((#1)^22)+1472062034232954033537024 ((#1)^24)-210795663759749098242048 ((#1)^26)+19553039013900542017536 ((#1)^28)-1065469437349025808384 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FEExpressionsNumericalApproximationElider]), Root[1-48 #1+104 ((#1)^2)-72 ((#1)^3)+16 ((#1)^4)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.219", FEExpressionsNumericalApproximationElider]), Root[44521-1506336 ((#1)^2)+13845632 ((#1)^4)-39671168 ((#1)^6)+39759360 ((#1)^8)-10698752 ((#1)^10)-1732608 ((#1)^12)+557056 ((#1)^14)+65536 ((#1)^16)&,6], ]),(NumericalApproximation[Root, (RowDefault["2.54", FEExpressionsNumericalApproximationElider]), Root[-389-5240 #1-16336 ((#1)^2)+9760 ((#1)^3)+19776 ((#1)^4)-17280 ((#1)^5)+5184 ((#1)^6)-1280 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["-1.13", FEExpressionsNumericalApproximationElider]), Root[211391026700294401-59952264805788373060 ((#1)^2)+1879593997492846567408 ((#1)^4)-24791183988969731655680 ((#1)^6)+176409087506817261627648 ((#1)^8)-761574600448175909923840 ((#1)^10)+2123820953027732914364416 ((#1)^12)-3952632026277989950668800 ((#1)^14)+«1»-4222671543983226991411200 «1»+2393894835212885233238016 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FEExpressionsNumericalApproximationElider]), Root[841-1159480 ((#1)^2)+2050704 ((#1)^4)+2225600 ((#1)^6)-4345344 ((#1)^8)-609280 ((#1)^10)+2207744 ((#1)^12)-737280 ((#1)^14)+65536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-0.0484", FEExpressionsNumericalApproximationElider]), Root[25+800 #1+6940 ((#1)^2)+24120 ((#1)^3)+37216 ((#1)^4)+24928 ((#1)^5)+7104 ((#1)^6)+1152 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["2.23", FEExpressionsNumericalApproximationElider]), Root[10462162449019571161-2159155264986839684636 ((#1)^2)+74160666927567635558016 ((#1)^4)-927758609593085932521472 ((#1)^6)+5842795857580150340071424 ((#1)^8)-21374700168613943938201600 ((#1)^10)+«6»+22355752856510619265794048 ((#1)^20)-6936955686385638105415680 ((#1)^22)+1472062034232954033537024 ((#1)^24)-210795663759749098242048 ((#1)^26)+19553039013900542017536 ((#1)^28)-1065469437349025808384 ((#1)^30)+26024157241122226176 ((#1)^32)&,24], ]),(NumericalApproximation[Root, 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((#1)^4)-32 ((#1)^6)+((#1)^8)&,7], ]),0},{(NumericalApproximation[Root, (RowDefault["2.25", FEExpressionsNumericalApproximationElider]), Root[1-92 ((#1)^2)+134 ((#1)^4)-28 ((#1)^6)+((#1)^8)&,7], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

{"Flower", 16}

GraphData[{"Flower",16},"Graph","MinimalCrossing"]

GraphData[{"Flower",16},"Graph","LCF"]

GraphData[{"Flower",16},"Graph","UnitDistance"]

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

GraphData[{"Flower",16},"Graph","All"]

GraphPlot[GraphData[{"Flower",16},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",16},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",16},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",16}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->5000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}]

v={{(NumericalApproximation[Root, (RowDefault["1.94", FEExpressionsNumericalApproximationElider]), Root[257-688 #1-64 ((#1)^2)+960 ((#1)^3)-960 ((#1)^4)+1280 ((#1)^5)+512 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.88", FEExpressionsNumericalApproximationElider]), Root[1-48 #1+704 ((#1)^2)-3264 ((#1)^3)+6208 ((#1)^4)-5376 ((#1)^5)+2560 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["1.05", FEExpressionsNumericalApproximationElider]), Root[1160321-89509808 #1+444019088 ((#1)^2)+3899434496 ((#1)^3)-4397630592 ((#1)^4)-34329076736 ((#1)^5)+19620380672 ((#1)^6)+105689884672 ((#1)^7)-71633523712 ((#1)^8)-127567118336 ((#1)^9)+120789696512 ((#1)^10)+42282319872 ((#1)^11)-71419428864 ((#1)^12)+5042601984 ((#1)^13)+18728091648 ((#1)^14)-7996440576 ((#1)^15)+1007681536 ((#1)^16)&,9], ]),(NumericalApproximation[Root, (RowDefault["2.34", FEExpressionsNumericalApproximationElider]), Root[13421377-166684672 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Root[-3215903-49543088 #1+38712064 ((#1)^2)+2545351168 ((#1)^3)+10177847488 ((#1)^4)+7884882944 ((#1)^5)-31043309568 ((#1)^6)-67515756544 ((#1)^7)-18912830464 ((#1)^8)+65688109056 ((#1)^9)+65414889472 ((#1)^10)+9293922304 ((#1)^11)-6514737152 ((#1)^12)+12793151488 ((#1)^13)+17314086912 ((#1)^14)+7151288320 ((#1)^15)+1007681536 ((#1)^16)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.0771", FEExpressionsNumericalApproximationElider]), Root[-3215903+49543088 #1+38712064 ((#1)^2)-2545351168 ((#1)^3)+10177847488 ((#1)^4)-7884882944 ((#1)^5)-31043309568 ((#1)^6)+67515756544 ((#1)^7)-18912830464 ((#1)^8)-65688109056 ((#1)^9)+65414889472 ((#1)^10)-9293922304 ((#1)^11)-6514737152 ((#1)^12)-12793151488 ((#1)^13)+17314086912 ((#1)^14)-7151288320 ((#1)^15)+1007681536 ((#1)^16)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,4], ])},{(1/2) (-2-(Sqrt[2])),(1/2) (2+(Sqrt[2]))},{(NumericalApproximation[Root, (RowDefault["-2.70", FEExpressionsNumericalApproximationElider]), Root[257+688 #1-64 ((#1)^2)-960 ((#1)^3)-960 ((#1)^4)-1280 ((#1)^5)+512 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.0426", FEExpressionsNumericalApproximationElider]), Root[1-48 #1+704 ((#1)^2)-3264 ((#1)^3)+6208 ((#1)^4)-5376 ((#1)^5)+2560 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-2.40", FEExpressionsNumericalApproximationElider]), Root[1160321+89509808 #1+444019088 ((#1)^2)-3899434496 ((#1)^3)-4397630592 ((#1)^4)+34329076736 ((#1)^5)+19620380672 ((#1)^6)-105689884672 ((#1)^7)-71633523712 ((#1)^8)+127567118336 ((#1)^9)+120789696512 ((#1)^10)-42282319872 ((#1)^11)-71419428864 ((#1)^12)-5042601984 ((#1)^13)+18728091648 ((#1)^14)+7996440576 ((#1)^15)+1007681536 ((#1)^16)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.909", FEExpressionsNumericalApproximationElider]), Root[13421377-166684672 #1+197518576 ((#1)^2)+1756827008 ((#1)^3)-3378780288 ((#1)^4)-4983136768 ((#1)^5)+13843644416 ((#1)^6)-28395520 ((#1)^7)-17494103040 ((#1)^8)+12042665984 ((#1)^9)-2005172224 ((#1)^10)-580845568 ((#1)^11)+3357671424 ((#1)^12)-6415187968 ((#1)^13)+7448559616 ((#1)^14)-4615831552 ((#1)^15)+1007681536 ((#1)^16)&,1], ])},{-1-(Sqrt[2]),1},{(NumericalApproximation[Root, (RowDefault["-2.23", FEExpressionsNumericalApproximationElider]), Root[1-40 ((#1)^2)+8 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-2.51", FEExpressionsNumericalApproximationElider]), Root[1-8 #1-24 ((#1)^2)+32 ((#1)^3)+16 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.995", FEExpressionsNumericalApproximationElider]), Root[-527+2224 ((#1)^2)-2720 ((#1)^4)+768 ((#1)^6)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.87", FEExpressionsNumericalApproximationElider]), Root[-19183-78192 #1+89152 ((#1)^2)+375104 ((#1)^3)-98656 ((#1)^4)-406016 ((#1)^5)-38400 ((#1)^6)+114688 ((#1)^7)+31744 ((#1)^8)&,2], ]),(NumericalApproximation[Root, (RowDefault["-1.76", FEExpressionsNumericalApproximationElider]), Root[1-71808 ((#1)^2)+35850560 ((#1)^4)-1070724096 ((#1)^6)+7383505920 ((#1)^8)-2188541952 ((#1)^10)-7990214656 ((#1)^12)-531628032 ((#1)^14)+1007681536 ((#1)^16)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-2.61", FEExpressionsNumericalApproximationElider]), Root[8-8 ((#1)^2)+((#1)^4)&,1], ]),0},{-1-(Sqrt[2]),0},{(NumericalApproximation[Root, (RowDefault["-1.94", FEExpressionsNumericalApproximationElider]), Root[257+688 #1-64 ((#1)^2)-960 ((#1)^3)-960 ((#1)^4)-1280 ((#1)^5)+512 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,2], ]),(NumericalApproximation[Root, (RowDefault["-1.88", FEExpressionsNumericalApproximationElider]), Root[1+48 #1+704 ((#1)^2)+3264 ((#1)^3)+6208 ((#1)^4)+5376 ((#1)^5)+2560 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.05", FEExpressionsNumericalApproximationElider]), Root[1160321+89509808 #1+444019088 ((#1)^2)-3899434496 ((#1)^3)-4397630592 ((#1)^4)+34329076736 ((#1)^5)+19620380672 ((#1)^6)-105689884672 ((#1)^7)-71633523712 ((#1)^8)+127567118336 ((#1)^9)+120789696512 ((#1)^10)-42282319872 ((#1)^11)-71419428864 ((#1)^12)-5042601984 ((#1)^13)+18728091648 ((#1)^14)+7996440576 ((#1)^15)+1007681536 ((#1)^16)&,4], ]),(NumericalApproximation[Root, (RowDefault["-2.34", FEExpressionsNumericalApproximationElider]), Root[13421377+166684672 #1+197518576 ((#1)^2)-1756827008 ((#1)^3)-3378780288 ((#1)^4)+4983136768 ((#1)^5)+13843644416 ((#1)^6)+28395520 ((#1)^7)-17494103040 ((#1)^8)-12042665984 ((#1)^9)-2005172224 ((#1)^10)+580845568 ((#1)^11)+3357671424 ((#1)^12)+6415187968 ((#1)^13)+7448559616 ((#1)^14)+4615831552 ((#1)^15)+1007681536 ((#1)^16)&,1], ])},{-1-(Sqrt[2]),-1},{(NumericalApproximation[Root, (RowDefault["-2.23", FEExpressionsNumericalApproximationElider]), Root[1-40 ((#1)^2)+8 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.07", FEExpressionsNumericalApproximationElider]), Root[49-112 #1-1696 ((#1)^2)-960 ((#1)^3)+2080 ((#1)^4)+2304 ((#1)^5)+1536 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,3], ]),(NumericalApproximation[Root, (RowDefault["-2.48", FEExpressionsNumericalApproximationElider]), Root[49-112 #1-1696 ((#1)^2)-960 ((#1)^3)+2080 ((#1)^4)+2304 ((#1)^5)+1536 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.0771", FEExpressionsNumericalApproximationElider]), Root[-3215903-49543088 #1+38712064 ((#1)^2)+2545351168 ((#1)^3)+10177847488 ((#1)^4)+7884882944 ((#1)^5)-31043309568 ((#1)^6)-67515756544 ((#1)^7)-18912830464 ((#1)^8)+65688109056 ((#1)^9)+65414889472 ((#1)^10)+9293922304 ((#1)^11)-6514737152 ((#1)^12)+12793151488 ((#1)^13)+17314086912 ((#1)^14)+7151288320 ((#1)^15)+1007681536 ((#1)^16)&,9], ]),(NumericalApproximation[Root, (RowDefault["-2.56", FEExpressionsNumericalApproximationElider]), Root[-3215903-49543088 #1+38712064 ((#1)^2)+2545351168 ((#1)^3)+10177847488 ((#1)^4)+7884882944 ((#1)^5)-31043309568 ((#1)^6)-67515756544 ((#1)^7)-18912830464 ((#1)^8)+65688109056 ((#1)^9)+65414889472 ((#1)^10)+9293922304 ((#1)^11)-6514737152 ((#1)^12)+12793151488 ((#1)^13)+17314086912 ((#1)^14)+7151288320 ((#1)^15)+1007681536 ((#1)^16)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["-1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,1], ])},{(1/2) (-2-(Sqrt[2])),(1/2) (-2-(Sqrt[2]))},{(NumericalApproximation[Root, (RowDefault["-0.0426", FEExpressionsNumericalApproximationElider]), Root[1+48 #1+704 ((#1)^2)+3264 ((#1)^3)+6208 ((#1)^4)+5376 ((#1)^5)+2560 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.70", FEExpressionsNumericalApproximationElider]), Root[257+688 #1-64 ((#1)^2)-960 ((#1)^3)-960 ((#1)^4)-1280 ((#1)^5)+512 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.909", FEExpressionsNumericalApproximationElider]), Root[13421377+166684672 #1+197518576 ((#1)^2)-1756827008 ((#1)^3)-3378780288 ((#1)^4)+4983136768 ((#1)^5)+13843644416 ((#1)^6)+28395520 ((#1)^7)-17494103040 ((#1)^8)-12042665984 ((#1)^9)-2005172224 ((#1)^10)+580845568 ((#1)^11)+3357671424 ((#1)^12)+6415187968 ((#1)^13)+7448559616 ((#1)^14)+4615831552 ((#1)^15)+1007681536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-2.40", FEExpressionsNumericalApproximationElider]), Root[1160321+89509808 #1+444019088 ((#1)^2)-3899434496 ((#1)^3)-4397630592 ((#1)^4)+34329076736 ((#1)^5)+19620380672 ((#1)^6)-105689884672 ((#1)^7)-71633523712 ((#1)^8)+127567118336 ((#1)^9)+120789696512 ((#1)^10)-42282319872 ((#1)^11)-71419428864 ((#1)^12)-5042601984 ((#1)^13)+18728091648 ((#1)^14)+7996440576 ((#1)^15)+1007681536 ((#1)^16)&,1], ])},{-1,-1-(Sqrt[2])},{(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ]),(NumericalApproximation[Root, (RowDefault["-2.23", FEExpressionsNumericalApproximationElider]), Root[1-40 ((#1)^2)+8 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.995", FEExpressionsNumericalApproximationElider]), Root[-527+2224 ((#1)^2)-2720 ((#1)^4)+768 ((#1)^6)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.51", FEExpressionsNumericalApproximationElider]), Root[1-8 #1-24 ((#1)^2)+32 ((#1)^3)+16 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.76", FEExpressionsNumericalApproximationElider]), Root[1-71808 ((#1)^2)+35850560 ((#1)^4)-1070724096 ((#1)^6)+7383505920 ((#1)^8)-2188541952 ((#1)^10)-7990214656 ((#1)^12)-531628032 ((#1)^14)+1007681536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-1.87", FEExpressionsNumericalApproximationElider]), Root[-19183-78192 #1+89152 ((#1)^2)+375104 ((#1)^3)-98656 ((#1)^4)-406016 ((#1)^5)-38400 ((#1)^6)+114688 ((#1)^7)+31744 ((#1)^8)&,2], ])},{0,(NumericalApproximation[Root, (RowDefault["-2.61", FEExpressionsNumericalApproximationElider]), Root[8-8 ((#1)^2)+((#1)^4)&,1], ])},{0,-1-(Sqrt[2])},{(NumericalApproximation[Root, (RowDefault["1.88", FEExpressionsNumericalApproximationElider]), Root[1-48 #1+704 ((#1)^2)-3264 ((#1)^3)+6208 ((#1)^4)-5376 ((#1)^5)+2560 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.94", FEExpressionsNumericalApproximationElider]), Root[257+688 #1-64 ((#1)^2)-960 ((#1)^3)-960 ((#1)^4)-1280 ((#1)^5)+512 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.34", FEExpressionsNumericalApproximationElider]), Root[13421377-166684672 #1+197518576 ((#1)^2)+1756827008 ((#1)^3)-3378780288 ((#1)^4)-4983136768 ((#1)^5)+13843644416 ((#1)^6)-28395520 ((#1)^7)-17494103040 ((#1)^8)+12042665984 ((#1)^9)-2005172224 ((#1)^10)-580845568 ((#1)^11)+3357671424 ((#1)^12)-6415187968 ((#1)^13)+7448559616 ((#1)^14)-4615831552 ((#1)^15)+1007681536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-1.05", FEExpressionsNumericalApproximationElider]), Root[1160321+89509808 #1+444019088 ((#1)^2)-3899434496 ((#1)^3)-4397630592 ((#1)^4)+34329076736 ((#1)^5)+19620380672 ((#1)^6)-105689884672 ((#1)^7)-71633523712 ((#1)^8)+127567118336 ((#1)^9)+120789696512 ((#1)^10)-42282319872 ((#1)^11)-71419428864 ((#1)^12)-5042601984 ((#1)^13)+18728091648 ((#1)^14)+7996440576 ((#1)^15)+1007681536 ((#1)^16)&,4], ])},{1,-1-(Sqrt[2])},{(NumericalApproximation[Root, (RowDefault["0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["-2.23", FEExpressionsNumericalApproximationElider]), Root[1-40 ((#1)^2)+8 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.48", FEExpressionsNumericalApproximationElider]), Root[49+112 #1-1696 ((#1)^2)+960 ((#1)^3)+2080 ((#1)^4)-2304 ((#1)^5)+1536 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.07", FEExpressionsNumericalApproximationElider]), Root[49-112 #1-1696 ((#1)^2)-960 ((#1)^3)+2080 ((#1)^4)+2304 ((#1)^5)+1536 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,3], ])},{(NumericalApproximation[Root, (RowDefault["2.56", FEExpressionsNumericalApproximationElider]), Root[-3215903+49543088 #1+38712064 ((#1)^2)-2545351168 ((#1)^3)+10177847488 ((#1)^4)-7884882944 ((#1)^5)-31043309568 ((#1)^6)+67515756544 ((#1)^7)-18912830464 ((#1)^8)-65688109056 ((#1)^9)+65414889472 ((#1)^10)-9293922304 ((#1)^11)-6514737152 ((#1)^12)-12793151488 ((#1)^13)+17314086912 ((#1)^14)-7151288320 ((#1)^15)+1007681536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-0.0771", FEExpressionsNumericalApproximationElider]), Root[-3215903-49543088 #1+38712064 ((#1)^2)+2545351168 ((#1)^3)+10177847488 ((#1)^4)+7884882944 ((#1)^5)-31043309568 ((#1)^6)-67515756544 ((#1)^7)-18912830464 ((#1)^8)+65688109056 ((#1)^9)+65414889472 ((#1)^10)+9293922304 ((#1)^11)-6514737152 ((#1)^12)+12793151488 ((#1)^13)+17314086912 ((#1)^14)+7151288320 ((#1)^15)+1007681536 ((#1)^16)&,9], ])},{(NumericalApproximation[Root, (RowDefault["1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["-1.85", FEExpressionsNumericalApproximationElider]), Root[2-4 ((#1)^2)+((#1)^4)&,1], ])},{(1/2) (2+(Sqrt[2])),(1/2) (-2-(Sqrt[2]))},{(NumericalApproximation[Root, (RowDefault["2.70", FEExpressionsNumericalApproximationElider]), Root[257-688 #1-64 ((#1)^2)+960 ((#1)^3)-960 ((#1)^4)+1280 ((#1)^5)+512 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-0.0426", FEExpressionsNumericalApproximationElider]), Root[1+48 #1+704 ((#1)^2)+3264 ((#1)^3)+6208 ((#1)^4)+5376 ((#1)^5)+2560 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["2.40", FEExpressionsNumericalApproximationElider]), Root[1160321-89509808 #1+444019088 ((#1)^2)+3899434496 ((#1)^3)-4397630592 ((#1)^4)-34329076736 ((#1)^5)+19620380672 ((#1)^6)+105689884672 ((#1)^7)-71633523712 ((#1)^8)-127567118336 ((#1)^9)+120789696512 ((#1)^10)+42282319872 ((#1)^11)-71419428864 ((#1)^12)+5042601984 ((#1)^13)+18728091648 ((#1)^14)-7996440576 ((#1)^15)+1007681536 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["0.909", FEExpressionsNumericalApproximationElider]), Root[13421377+166684672 #1+197518576 ((#1)^2)-1756827008 ((#1)^3)-3378780288 ((#1)^4)+4983136768 ((#1)^5)+13843644416 ((#1)^6)+28395520 ((#1)^7)-17494103040 ((#1)^8)-12042665984 ((#1)^9)-2005172224 ((#1)^10)+580845568 ((#1)^11)+3357671424 ((#1)^12)+6415187968 ((#1)^13)+7448559616 ((#1)^14)+4615831552 ((#1)^15)+1007681536 ((#1)^16)&,12], ])},{1+(Sqrt[2]),-1},{(NumericalApproximation[Root, (RowDefault["2.23", FEExpressionsNumericalApproximationElider]), Root[1-40 ((#1)^2)+8 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["-0.924", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.51", FEExpressionsNumericalApproximationElider]), Root[1+8 #1-24 ((#1)^2)-32 ((#1)^3)+16 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.995", FEExpressionsNumericalApproximationElider]), Root[-527+2224 ((#1)^2)-2720 ((#1)^4)+768 ((#1)^6)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["1.87", FEExpressionsNumericalApproximationElider]), Root[-19183+78192 #1+89152 ((#1)^2)-375104 ((#1)^3)-98656 ((#1)^4)+406016 ((#1)^5)-38400 ((#1)^6)-114688 ((#1)^7)+31744 ((#1)^8)&,7], ]),(NumericalApproximation[Root, (RowDefault["1.76", FEExpressionsNumericalApproximationElider]), Root[1-71808 ((#1)^2)+35850560 ((#1)^4)-1070724096 ((#1)^6)+7383505920 ((#1)^8)-2188541952 ((#1)^10)-7990214656 ((#1)^12)-531628032 ((#1)^14)+1007681536 ((#1)^16)&,12], ])},{(NumericalApproximation[Root, (RowDefault["2.61", FEExpressionsNumericalApproximationElider]), Root[8-8 ((#1)^2)+((#1)^4)&,4], ]),0},{1+(Sqrt[2]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

{"Flower", 17}

GraphData[{"Flower",17},"Graph","MinimalCrossing"]

GraphData[{"Flower",17},"Graph","LCF"]

GraphData[{"Flower",17},"Graph","UnitDistance"]

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

GraphData[{"Flower",17},"Graph","All"]

GraphPlot[GraphData[{"Flower",17},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",17},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",17},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",17}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->5000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}];//Timing

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((#1)^25)+20654023013325340672 ((#1)^26)-2595733240840978432 ((#1)^27)-1223935264020234240 ((#1)^28)+783656022513287168 ((#1)^29)+495419469661732864 ((#1)^30)+103487855472410624 ((#1)^31)+7955658116694016 ((#1)^32)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.758", FEExpressionsNumericalApproximationElider]), Root[1-85 ((#1)^2)+799 ((#1)^4)-2771 ((#1)^6)+4590 ((#1)^8)-3893 ((#1)^10)+1649 ((#1)^12)-306 ((#1)^14)+17 ((#1)^16)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.66", FEExpressionsNumericalApproximationElider]), Root[-1-5 #1+5 ((#1)^2)+19 ((#1)^3)-8 ((#1)^4)-21 ((#1)^5)+3 ((#1)^6)+6 ((#1)^7)+((#1)^8)&,2], ])},{(NumericalApproximation[Root, (RowDefault["-0.706", FEExpressionsNumericalApproximationElider]), Root[1-816 ((#1)^2)+139808 ((#1)^4)-3018112 ((#1)^6)+21216000 ((#1)^8)-52398080 ((#1)^10)+49508352 ((#1)^12)-15876096 ((#1)^14)+1114112 ((#1)^16)&,4], ]),(NumericalApproximation[Root, (RowDefault["-2.48", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1+192 ((#1)^2)-1120 ((#1)^3)+944 ((#1)^4)+2240 ((#1)^5)-2496 ((#1)^6)-640 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.24", FEExpressionsNumericalApproximationElider]), Root[4679517649-364360218000 ((#1)^2)+10988481854656 ((#1)^4)-159602466575872 ((#1)^6)+1116332814981120 ((#1)^8)-4104257673693184 ((#1)^10)+8081452888727552 ((#1)^12)-7167891995295744 ((#1)^14)-1474456523767808 ((#1)^16)+8618963097616384 ((#1)^18)-5691970560196608 ((#1)^20)-854962418483200 ((#1)^22)+2460900212604928 ((#1)^24)-878491536982016 ((#1)^26)+18773838921728 ((#1)^28)+23893976809472 ((#1)^30)+1241245548544 ((#1)^32)&,19], ]),(NumericalApproximation[Root, (RowDefault["-2.57", FEExpressionsNumericalApproximationElider]), Root[103+27100 #1+274144 ((#1)^2)+724128 ((#1)^3)-2361632 ((#1)^4)-4264960 ((#1)^5)+5333504 ((#1)^6)+3794176 ((#1)^7)-1828352 ((#1)^8)+8271360 ((#1)^9)-4361216 ((#1)^10)-12011520 ((#1)^11)+1216512 ((#1)^12)+3768320 ((#1)^13)+1081344 ((#1)^14)+294912 ((#1)^15)+65536 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((#1)^27)+737475029029093376 ((#1)^28)-167909770215292928 ((#1)^29)-2719591545438208 ((#1)^30)+40497280993722368 ((#1)^31)+7955658116694016 ((#1)^32)&,4], ])},{(NumericalApproximation[Root, (RowDefault["0.255", FEExpressionsNumericalApproximationElider]), Root[1-34 ((#1)^2)+391 ((#1)^4)-1836 ((#1)^6)+3757 ((#1)^8)-3740 ((#1)^10)+1819 ((#1)^12)-374 ((#1)^14)+17 ((#1)^16)&,9], ]),(NumericalApproximation[Root, (RowDefault["-2.76", FEExpressionsNumericalApproximationElider]), Root[1+2 #1-11 ((#1)^2)-14 ((#1)^3)+19 ((#1)^4)+14 ((#1)^5)-11 ((#1)^6)-2 ((#1)^7)+((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.238", FEExpressionsNumericalApproximationElider]), Root[1-272 ((#1)^2)+24480 ((#1)^4)-885632 ((#1)^6)+11484928 ((#1)^8)-39446528 ((#1)^10)+48115712 ((#1)^12)-20332544 ((#1)^14)+1114112 ((#1)^16)&,12], ]),(NumericalApproximation[Root, (RowDefault["-2.57", FEExpressionsNumericalApproximationElider]), Root[1-32 #1+40 ((#1)^2)+768 ((#1)^3)-1936 ((#1)^4)-768 ((#1)^5)+2624 ((#1)^6)+1664 ((#1)^7)+256 ((#1)^8)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.08", FEExpressionsNumericalApproximationElider]), Root[139129-5330827632 ((#1)^2)+390333359552 ((#1)^4)-11421498505216 ((#1)^6)+168310095473664 ((#1)^8)-1302232912017408 ((#1)^10)+5184647642095616 ((#1)^12)-10542743953408000 ((#1)^14)+9423857730584576 ((#1)^16)+219308406079488 ((#1)^18)-5671220910489600 ((#1)^20)+2010763241717760 ((#1)^22)+934381526974464 ((#1)^24)-382536362491904 ((#1)^26)-44529684054016 ((#1)^28)+11481521324032 ((#1)^30)+1241245548544 ((#1)^32)&,20], ]),(NumericalApproximation[Root, (RowDefault["-1.95", FEExpressionsNumericalApproximationElider]), Root[137+2828 #1+26048 ((#1)^2)+97152 ((#1)^3)-71264 ((#1)^4)-1887168 ((#1)^5)-4803328 ((#1)^6)+2584064 ((#1)^7)+19645440 ((#1)^8)+6503936 ((#1)^9)-23843840 ((#1)^10)-14174208 ((#1)^11)+6615040 ((#1)^12)+6569984 ((#1)^13)+1736704 ((#1)^14)+360448 ((#1)^15)+65536 ((#1)^16)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.51", FEExpressionsNumericalApproximationElider]), Root[697571869461359823443661441049-2252209103266296022177512366505408 ((#1)^2)+363879974237984412571637791405504352 ((#1)^4)-21690907267574172232357367830029584576 ((#1)^6)+«36»+339873100373801769167002368788134291832832 ((#1)^56)-39941108296126212606397652216944126853120 ((#1)^58)+3325279925413628092070647159847064174592 ((#1)^60)-181881284079984037710633376272231694336 ((#1)^62)+5286252564239032128390425949163749376 ((#1)^64)&,40], ]),(NumericalApproximation[Root, (RowDefault["-1.05", FEExpressionsNumericalApproximationElider]), Root[245921044339423-814208172407052 #1-25490550650065744 ((#1)^2)+103294805531337352 ((#1)^3)+620075383668457872 ((#1)^4)-2638658988343576256 ((#1)^5)-5327681895805211776 ((#1)^6)+22867116380319969664 ((#1)^7)+«26»+5021220410587348992 ((#1)^25)+9697837764191453184 ((#1)^26)+4789524855651303424 ((#1)^27)+1522962431502974976 ((#1)^28)+566650934256467968 ((#1)^29)+244968220698607616 ((#1)^30)+68979238505545728 ((#1)^31)+7955658116694016 ((#1)^32)&,7], ])},{(NumericalApproximation[Root, (RowDefault["1.23", FEExpressionsNumericalApproximationElider]), Root[1-51 ((#1)^2)+680 ((#1)^4)-2737 ((#1)^6)+5015 ((#1)^8)-4709 ((#1)^10)+2244 ((#1)^12)-459 ((#1)^14)+17 ((#1)^16)&,15], ]),(NumericalApproximation[Root, (RowDefault["-2.48", FEExpressionsNumericalApproximationElider]), Root[-1+3 #1+12 ((#1)^2)-9 ((#1)^3)-25 ((#1)^4)-((#1)^5)+14 ((#1)^6)+7 ((#1)^7)+((#1)^8)&,2], ])},{(NumericalApproximation[Root, (RowDefault["1.15", FEExpressionsNumericalApproximationElider]), Root[1-1632 ((#1)^2)+121312 ((#1)^4)-2617728 ((#1)^6)+17891072 ((#1)^8)-50657280 ((#1)^10)+61485056 ((#1)^12)-25903104 ((#1)^14)+1114112 ((#1)^16)&,15], ]),(NumericalApproximation[Root, (RowDefault["-2.31", FEExpressionsNumericalApproximationElider]), Root[-1+20 #1-56 ((#1)^2)-672 ((#1)^3)+2032 ((#1)^4)+1920 ((#1)^5)-1856 ((#1)^6)-384 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.65", FEExpressionsNumericalApproximationElider]), Root[1666681-286090144 ((#1)^2)+5445373632 ((#1)^4)-2058041490176 ((#1)^6)+53669615451136 ((#1)^8)-460042796263424 ((#1)^10)+1594788116209664 ((#1)^12)-2261603868409856 ((#1)^14)+388288186286080 ((#1)^16)+1953821784014848 ((#1)^18)-1199501697089536 ((#1)^20)-499243747901440 ((#1)^22)+444128324222976 ((#1)^24)+45383040368640 ((#1)^26)-50735911796736 ((#1)^28)-3413425258496 ((#1)^30)+1241245548544 ((#1)^32)&,20], ]),(NumericalApproximation[Root, (RowDefault["-1.07", FEExpressionsNumericalApproximationElider]), Root[4283+41492 #1-308440 ((#1)^2)-535504 ((#1)^3)+2451584 ((#1)^4)+3913280 ((#1)^5)+631808 ((#1)^6)+2236672 ((#1)^7)-4214272 ((#1)^8)-23920128 ((#1)^9)-17867776 ((#1)^10)+12269568 ((#1)^11)+22122496 ((#1)^12)+10977280 ((#1)^13)+2523136 ((#1)^14)+425984 ((#1)^15)+65536 ((#1)^16)&,5], ])},{(NumericalApproximation[Root, (RowDefault["2.72", FEExpressionsNumericalApproximationElider]), Root[320322967034179912283085297190369-60578964645335168790016570064980672 ((#1)^2)+4278825184686756443441042987137011936 ((#1)^4)-151215522222600914824924470266732598912 ((#1)^6)+«35»+170673374022997217596214409693121885503488 ((#1)^56)-31007533764072382913576074005412849909760 ((#1)^58)+3270969468002092219664438397730247671808 ((#1)^60)-195544095766664753071815725697583808512 ((#1)^62)+5286252564239032128390425949163749376 ((#1)^64)&,40], ]),(NumericalApproximation[Root, (RowDefault["-0.0689", FEExpressionsNumericalApproximationElider]), Root[-15092832509+4669127876116 #1+252939901286920 ((#1)^2)+4092466207353504 ((#1)^3)+«31»+568503591195035828224 ((#1)^24)+350902051067695988736 ((#1)^25)+164730641304550113280 ((#1)^26)+63638146262287515648 ((#1)^27)+20499342083898736640 ((#1)^28)+5321887204920786944 ((#1)^29)+1033104464795402240 ((#1)^30)+130900095544590336 ((#1)^31)+7955658116694016 ((#1)^32)&,14], ])},{(NumericalApproximation[Root, (RowDefault["2.05", FEExpressionsNumericalApproximationElider]), Root[1-68 ((#1)^2)+850 ((#1)^4)-3706 ((#1)^6)+7463 ((#1)^8)-7446 ((#1)^10)+3451 ((#1)^12)-561 ((#1)^14)+17 ((#1)^16)&,15], ]),(NumericalApproximation[Root, (RowDefault["-1.86", FEExpressionsNumericalApproximationElider]), Root[1+4 #1-10 ((#1)^2)-10 ((#1)^3)+15 ((#1)^4)+6 ((#1)^5)-7 ((#1)^6)-((#1)^7)+((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.91", FEExpressionsNumericalApproximationElider]), Root[1-272 ((#1)^2)+24480 ((#1)^4)-850816 ((#1)^6)+12773120 ((#1)^8)-77395968 ((#1)^10)+124153856 ((#1)^12)-32587776 ((#1)^14)+1114112 ((#1)^16)&,15], ]),(NumericalApproximation[Root, (RowDefault["-1.74", FEExpressionsNumericalApproximationElider]), Root[1+72 #1+840 ((#1)^2)+3696 ((#1)^3)+7920 ((#1)^4)+9152 ((#1)^5)+5824 ((#1)^6)+1920 ((#1)^7)+256 ((#1)^8)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.85", FEExpressionsNumericalApproximationElider]), Root[142396489-9001046576 ((#1)^2)+314734625024 ((#1)^4)-3976139842048 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((#1)^4)-12376 ((#1)^6)+24310 ((#1)^8)-19448 ((#1)^10)+6188 ((#1)^12)-680 ((#1)^14)+17 ((#1)^16)&,15], ]),-1},{(NumericalApproximation[Root, (RowDefault["2.41", FEExpressionsNumericalApproximationElider]), Root[1-13872 ((#1)^2)+2635680 ((#1)^4)-63959168 ((#1)^6)+355641088 ((#1)^8)-572549120 ((#1)^10)+285143040 ((#1)^12)-40386560 ((#1)^14)+1114112 ((#1)^16)&,15], ]),(NumericalApproximation[Root, (RowDefault["-0.932", FEExpressionsNumericalApproximationElider]), Root[1+8 #1-40 ((#1)^2)-80 ((#1)^3)+240 ((#1)^4)+192 ((#1)^5)-448 ((#1)^6)-128 ((#1)^7)+256 ((#1)^8)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.67", FEExpressionsNumericalApproximationElider]), Root[1-544 ((#1)^2)+38080 ((#1)^4)-792064 ((#1)^6)+6223360 ((#1)^8)-19914752 ((#1)^10)+25346048 ((#1)^12)-11141120 ((#1)^14)+1114112 ((#1)^16)&,16], ]),(NumericalApproximation[Root, (RowDefault["0.996", FEExpressionsNumericalApproximationElider]), Root[-12239-1407200 ((#1)^2)+8244416 ((#1)^4)-17570304 ((#1)^6)+15685120 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((#1)^14)-10576694931347996672 ((#1)^16)-5705236318289920000 ((#1)^18)+15033358552909479936 ((#1)^20)+7118739002663895040 ((#1)^22)-5266086613045215232 ((#1)^24)-1772140161459552256 ((#1)^26)+290028265031073792 ((#1)^28)+115774996697055232 ((#1)^30)+7955658116694016 ((#1)^32)&,20], ])},{(NumericalApproximation[Root, (RowDefault["2.77", FEExpressionsNumericalApproximationElider]), Root[65536-278528 ((#1)^2)+487424 ((#1)^4)-452608 ((#1)^6)+239360 ((#1)^8)-71808 ((#1)^10)+11424 ((#1)^12)-816 ((#1)^14)+17 ((#1)^16)&,15], ]),0},{(NumericalApproximation[Root, (RowDefault["2.58", FEExpressionsNumericalApproximationElider]), Root[1-136 ((#1)^2)+2380 ((#1)^4)-12376 ((#1)^6)+24310 ((#1)^8)-19448 ((#1)^10)+6188 ((#1)^12)-680 ((#1)^14)+17 ((#1)^16)&,15], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

{"Flower", 18}

GraphData[{"Flower",18},"Graph","MinimalCrossing"]

GraphData[{"Flower",18},"Graph","LCF"]

GraphData[{"Flower",18},"Graph","UnitDistance"]

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

GraphData[{"Flower",18},"Graph","All"]

GraphPlot[GraphData[{"Flower",18},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",18},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",18},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",18}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->2000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}];//Timing

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(RowDefault["-2.85", FEExpressionsNumericalApproximationElider]), Root[9192457+32990454 #1-93936084 ((#1)^2)-249504624 ((#1)^3)+231750000 ((#1)^4)+667191168 ((#1)^5)-99254208 ((#1)^6)-820146816 ((#1)^7)-257925888 ((#1)^8)+448851456 ((#1)^9)+400536576 ((#1)^10)+122923008 ((#1)^11)+13307904 ((#1)^12)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.46", FEExpressionsNumericalApproximationElider]), Root[-1+6 ((#1)^2)-9 ((#1)^4)+3 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-2.53", FEExpressionsNumericalApproximationElider]), Root[-3+3 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-1.37", FEExpressionsNumericalApproximationElider]), Root[-1+132 ((#1)^2)-432 ((#1)^4)+192 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-2.38", FEExpressionsNumericalApproximationElider]), Root[-3-18 #1+12 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.489", FEExpressionsNumericalApproximationElider]), Root[1-24 ((#1)^2)-48 ((#1)^4)-640 ((#1)^6)+7680 ((#1)^8)-12288 ((#1)^10)+4096 ((#1)^12)&,6], ]),(NumericalApproximation[Root, (RowDefault["-2.97", FEExpressionsNumericalApproximationElider]), Root[-359+432 #1+1068 ((#1)^2)+1152 ((#1)^3)+2016 ((#1)^4)+1152 ((#1)^5)+192 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.39", FEExpressionsNumericalApproximationElider]), Root[26162092009-2547777020748 ((#1)^2)+53907539119776 ((#1)^4)-528198462884352 ((#1)^6)+2944591189684992 ((#1)^8)-10219793165899776 ((#1)^10)+23102069112225792 ((#1)^12)-34729052493201408 ((#1)^14)+34759482625622016 ((#1)^16)-22675936151863296 ((#1)^18)+9169256768864256 ((#1)^20)-2029172167802880 ((#1)^22)+177100308873216 ((#1)^24)&,15], ]),(NumericalApproximation[Root, (RowDefault["-2.52", FEExpressionsNumericalApproximationElider]), Root[110953-26103510 #1-163449024 ((#1)^2)-286690032 ((#1)^3)-132766704 ((#1)^4)+306036864 ((#1)^5)+859816320 ((#1)^6)+861252480 ((#1)^7)+905384448 ((#1)^8)+909121536 ((#1)^9)+505598976 ((#1)^10)+133429248 ((#1)^11)+13307904 ((#1)^12)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.508", FEExpressionsNumericalApproximationElider]), Root[-1+6 ((#1)^2)-9 ((#1)^4)+3 ((#1)^6)&,3], ]),(NumericalApproximation[Root, (RowDefault["-2.88", FEExpressionsNumericalApproximationElider]), Root[-1+3 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.477", FEExpressionsNumericalApproximationElider]), Root[-1+60 ((#1)^2)-288 ((#1)^4)+192 ((#1)^6)&,2], ]),(NumericalApproximation[Root, (RowDefault["-2.71", FEExpressionsNumericalApproximationElider]), Root[-1+6 #1+24 ((#1)^2)+8 ((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.47", FEExpressionsNumericalApproximationElider]), Root[1-24 ((#1)^2)-48 ((#1)^4)-640 ((#1)^6)+7680 ((#1)^8)-12288 ((#1)^10)+4096 ((#1)^12)&,8], ]),(NumericalApproximation[Root, (RowDefault["-2.62", FEExpressionsNumericalApproximationElider]), Root[-359+432 #1+1068 ((#1)^2)+1152 ((#1)^3)+2016 ((#1)^4)+1152 ((#1)^5)+192 ((#1)^6)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.17", FEExpressionsNumericalApproximationElider]), Root[26162092009-2547777020748 ((#1)^2)+53907539119776 ((#1)^4)-528198462884352 ((#1)^6)+2944591189684992 ((#1)^8)-10219793165899776 ((#1)^10)+23102069112225792 ((#1)^12)-34729052493201408 ((#1)^14)+34759482625622016 ((#1)^16)-22675936151863296 ((#1)^18)+9169256768864256 ((#1)^20)-2029172167802880 ((#1)^22)+177100308873216 ((#1)^24)&,16], ]),(NumericalApproximation[Root, (RowDefault["-1.90", FEExpressionsNumericalApproximationElider]), Root[110953-26103510 #1-163449024 ((#1)^2)-286690032 ((#1)^3)-132766704 ((#1)^4)+306036864 ((#1)^5)+859816320 ((#1)^6)+861252480 ((#1)^7)+905384448 ((#1)^8)+909121536 ((#1)^9)+505598976 ((#1)^10)+133429248 ((#1)^11)+13307904 ((#1)^12)&,4], ])},{(NumericalApproximation[Root, (RowDefault["0.508", FEExpressionsNumericalApproximationElider]), Root[-1+6 ((#1)^2)-9 ((#1)^4)+3 ((#1)^6)&,4], ]),(NumericalApproximation[Root, (RowDefault["-2.88", 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g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

{"Flower", 19}

GraphData[{"Flower",19},"Graph","MinimalCrossing"]

GraphData[{"Flower",19},"Graph","LCF"]

GraphData[{"Flower",19},"Graph","UnitDistance"]

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

GraphData[{"Flower",19},"Graph","All"]

GraphPlot[GraphData[{"Flower",19},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",19},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",19},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",19}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->72,WorkingPrecision->5000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}];//Timing

v={{(NumericalApproximation[Root, (RowDefault["2.51", FEExpressionsNumericalApproximationElider]), Root[98282877001-1446052314336 ((#1)^2)+1580807157984 ((#1)^4)+25940660096960 ((#1)^6)+95563782849792 ((#1)^8)-1204687882615808 ((#1)^10)+1938907957899264 ((#1)^12)+4596802098905088 ((#1)^14)-13049288140193792 ((#1)^16)-3390969591889920 ((#1)^18)+31057270070575104 ((#1)^20)-12770825857925120 ((#1)^22)-28522561646100480 ((#1)^24)+23727868009775104 ((#1)^26)+4786147809034240 ((#1)^28)-10719265560723456 ((#1)^30)+3919621514067968 ((#1)^32)-539568151461888 ((#1)^34)+24807731101696 ((#1)^36)&,23], ]),(NumericalApproximation[Root, (RowDefault["1.92", FEExpressionsNumericalApproximationElider]), Root[19-1596 #1+50008 ((#1)^2)-771704 ((#1)^3)+6920560 ((#1)^4)-40291552 ((#1)^5)+163796416 ((#1)^6)-485988992 ((#1)^7)+1075123968 ((#1)^8)-1783405568 ((#1)^9)+2217971712 ((#1)^10)-2077915136 ((#1)^11)+1488121856 ((#1)^12)-824115200 ((#1)^13)+343572480 ((#1)^14)-98959360 ((#1)^15)+17825792 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((#1)^70)+41463320838031668759437175441538976428589056 ((#1)^72)&,48], ]),(NumericalApproximation[Root, (RowDefault["0.933", FEExpressionsNumericalApproximationElider]), Root[-329815745698495373+820815199814774028 #1+16034285237687833200 ((#1)^2)-10077733778638310160 ((#1)^3)-279364493595933995504 ((#1)^4)-62808122604612127264 ((#1)^5)+2519466240184404447360 ((#1)^6)+«33»+111073070198760597880832 ((#1)^29)+84809091723503983394816 ((#1)^30)+42806329687743211241472 ((#1)^31)+16440305194785652604928 ((#1)^32)+4863645764246803841024 ((#1)^33)+1093875414881485717504 ((#1)^34)+183927422713579175936 ((#1)^35)+17837124318455136256 ((#1)^36)&,24], ])},{(NumericalApproximation[Root, (RowDefault["2.91", FEExpressionsNumericalApproximationElider]), Root[-1+171 ((#1)^2)-3876 ((#1)^4)+27132 ((#1)^6)-75582 ((#1)^8)+92378 ((#1)^10)-50388 ((#1)^12)+11628 ((#1)^14)-969 ((#1)^16)+19 ((#1)^18)&,17], ]),-1},{(NumericalApproximation[Root, (RowDefault["2.76", FEExpressionsNumericalApproximationElider]), Root[-1+21660 ((#1)^2)-6480672 ((#1)^4)+252942592 ((#1)^6)-2348752640 ((#1)^8)+6719032320 ((#1)^10)-6601420800 ((#1)^12)+2238840832 ((#1)^14)-232849408 ((#1)^16)+4980736 ((#1)^18)&,17], ]),(NumericalApproximation[Root, (RowDefault["-0.946", FEExpressionsNumericalApproximationElider]), Root[-1+10 #1+40 ((#1)^2)-160 ((#1)^3)-240 ((#1)^4)+672 ((#1)^5)+448 ((#1)^6)-1024 ((#1)^7)-256 ((#1)^8)+512 ((#1)^9)&,1], ])},{(NumericalApproximation[Root, (RowDefault["3.00", FEExpressionsNumericalApproximationElider]), Root[-1+684 ((#1)^2)-62016 ((#1)^4)+1736448 ((#1)^6)-19348992 ((#1)^8)+94595072 ((#1)^10)-206389248 ((#1)^12)+190513152 ((#1)^14)-63504384 ((#1)^16)+4980736 ((#1)^18)&,18], ]),(NumericalApproximation[Root, (RowDefault["0.997", FEExpressionsNumericalApproximationElider]), Root[1782959-7868252 ((#1)^2)-3808320 ((#1)^4)+77036800 ((#1)^6)-167558656 ((#1)^8)+149649408 ((#1)^10)-41631744 ((#1)^12)-16711680 ((#1)^14)+8847360 ((#1)^16)+262144 ((#1)^18)&,12], ])},{(NumericalApproximation[Root, (RowDefault["2.43", FEExpressionsNumericalApproximationElider]), Root[249566204697206108089-55695344572369667890480 ((#1)^2)+1663533366453141592054688 ((#1)^4)-22742455741971894919291136 ((#1)^6)+181611119674111345504873216 ((#1)^8)-939629952615122353350220800 ((#1)^10)+«9»+5648313087721905630065197056 ((#1)^24)-1874416682524380019633946624 ((#1)^26)+423411569131410183860256768 ((#1)^28)-62756111654337548090605568 ((#1)^30)+5768260818843761209507840 ((#1)^32)-295823796336660909654016 ((#1)^34)+6439201878962304188416 ((#1)^36)&,33], ]),(NumericalApproximation[Root, (RowDefault["1.82", FEExpressionsNumericalApproximationElider]), Root[361-55599776 ((#1)^2)+487499748576 ((#1)^4)-627842155708160 ((#1)^6)+166924071427964672 ((#1)^8)-11468898095349376000 ((#1)^10)+263073325618001534976 ((#1)^12)-1692038764616262909952 ((#1)^14)-998713086814592761856 ((#1)^16)+«1»-16688176026425206243328 ((#1)^20)-53092492092751701082112 ((#1)^22)+23717743457123006676992 ((#1)^24)+34786523459675052048384 ((#1)^26)-7654979702607073771520 ((#1)^28)-6560083085747699056640 ((#1)^30)+523870679906411085824 ((#1)^32)+284342769993883058176 ((#1)^34)+17837124318455136256 ((#1)^36)&,24], ])},{(NumericalApproximation[Root, (RowDefault["3.08", FEExpressionsNumericalApproximationElider]), Root[-262144+1245184 ((#1)^2)-2490368 ((#1)^4)+2723840 ((#1)^6)-1770496 ((#1)^8)+695552 ((#1)^10)-160512 ((#1)^12)+20064 ((#1)^14)-1140 ((#1)^16)+19 ((#1)^18)&,17], ]),0},{(NumericalApproximation[Root, (RowDefault["2.91", FEExpressionsNumericalApproximationElider]), Root[-1+171 ((#1)^2)-3876 ((#1)^4)+27132 ((#1)^6)-75582 ((#1)^8)+92378 ((#1)^10)-50388 ((#1)^12)+11628 ((#1)^14)-969 ((#1)^16)+19 ((#1)^18)&,17], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

{"Flower", 20}

GraphData[{"Flower",20},"Graph","MinimalCrossing"]

GraphData[{"Flower",20},"Graph","LCF"]

GraphData[{"Flower",20},"Graph","UnitDistance"]

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

GraphData[{"Flower",20},"Graph","All"]

GraphPlot[GraphData[{"Flower",20},"EdgeList"]]

gs=Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"Flower",20},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

Graph[GraphData[{"Flower",20},"Edges"]]

gs=Select[{UnitDistanceEmbeddingRefine@%},UnitDistanceGraphEmbeddingQ]

Select[FindUnitDistanceEmbedding[GraphData[{"Flower",20}],20],UnitDistanceGraphEmbeddingQ]

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->2000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}];//Timing

v={{(NumericalApproximation[Root, (RowDefault["2.69", FEExpressionsNumericalApproximationElider]), Root[1+160 #1-1312 ((#1)^2)+2480 ((#1)^3)-416 ((#1)^4)+1920 ((#1)^5)+832 ((#1)^6)-1280 ((#1)^7)+256 ((#1)^8)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.92", FEExpressionsNumericalApproximationElider]), Root[1-56 #1+832 ((#1)^2)-4608 ((#1)^3)+10400 ((#1)^4)-8832 ((#1)^5)+3392 ((#1)^6)-1024 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["1.92", FEExpressionsNumericalApproximationElider]), Root[-1458650399-5312973220 #1-4264999300 ((#1)^2)+3261379040 ((#1)^3)+6856485744 ((#1)^4)+12330873920 ((#1)^5)+9901261760 ((#1)^6)-16529662720 ((#1)^7)-18399839744 ((#1)^8)+11355130880 ((#1)^9)+11007411200 ((#1)^10)-5310545920 ((#1)^11)-2778853376 ((#1)^12)+1443676160 ((#1)^13)+219545600 ((#1)^14)-186777600 ((#1)^15)+23658496 ((#1)^16)&,5], ]),(NumericalApproximation[Root, (RowDefault["2.56", FEExpressionsNumericalApproximationElider]), Root[58917025-26321800 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((#1)^12)+17203200 ((#1)^14)+23658496 ((#1)^16)&,1], ]),(NumericalApproximation[Root, (RowDefault["2.62", FEExpressionsNumericalApproximationElider]), Root[895-94480 #1+165620 ((#1)^2)-2400 ((#1)^3)-194416 ((#1)^4)+96576 ((#1)^5)+35264 ((#1)^6)-30464 ((#1)^7)+4864 ((#1)^8)&,5], ])},{0,1+(Sqrt[5])},{0,(NumericalApproximation[Root, (RowDefault["3.08", FEExpressionsNumericalApproximationElider]), Root[5-10 ((#1)^2)+((#1)^4)&,4], ])},{(NumericalApproximation[Root, (RowDefault["-1.92", FEExpressionsNumericalApproximationElider]), Root[1+56 #1+832 ((#1)^2)+4608 ((#1)^3)+10400 ((#1)^4)+8832 ((#1)^5)+3392 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,1], ]),(NumericalApproximation[Root, (RowDefault["2.69", FEExpressionsNumericalApproximationElider]), Root[1+160 #1-1312 ((#1)^2)+2480 ((#1)^3)-416 ((#1)^4)+1920 ((#1)^5)+832 ((#1)^6)-1280 ((#1)^7)+256 ((#1)^8)&,5], ])},{(NumericalApproximation[Root, (RowDefault["-2.56", FEExpressionsNumericalApproximationElider]), Root[58917025+26321800 #1-262782100 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((#1)^7)+256 ((#1)^8)&,3], ])},{(NumericalApproximation[Root, (RowDefault["3.20", FEExpressionsNumericalApproximationElider]), Root[-208690975+1449127100 #1-2922216800 ((#1)^2)+385626400 ((#1)^3)+4668683520 ((#1)^4)-4881602240 ((#1)^5)-3887128320 ((#1)^6)+11544391680 ((#1)^7)-3126694144 ((#1)^8)-8422934528 ((#1)^9)+5749403648 ((#1)^10)+1233956864 ((#1)^11)-1752125440 ((#1)^12)-296534016 ((#1)^13)+659406848 ((#1)^14)-220397568 ((#1)^15)+23658496 ((#1)^16)&,8], ]),(NumericalApproximation[Root, (RowDefault["-0.0507", FEExpressionsNumericalApproximationElider]), Root[-5961359-161461780 #1-961749840 ((#1)^2)-1865769120 ((#1)^3)+1996931584 ((#1)^4)+17786248640 ((#1)^5)+43441949440 ((#1)^6)+62650563840 ((#1)^7)+61180439296 ((#1)^8)+43950049280 ((#1)^9)+26248376320 ((#1)^10)+15227002880 ((#1)^11)+8702767104 ((#1)^12)+4151214080 ((#1)^13)+1385349120 ((#1)^14)+273940480 ((#1)^15)+23658496 ((#1)^16)&,7], ])},{(1/2) (3+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-1.90", FEExpressionsNumericalApproximationElider]), Root[5-5 ((#1)^2)+((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["2.49", FEExpressionsNumericalApproximationElider]), Root[5-100 ((#1)^2)+16 ((#1)^4)&,4], ]),(1/4) (-5-(Sqrt[5]))},{(NumericalApproximation[Root, (RowDefault["3.31", FEExpressionsNumericalApproximationElider]), Root[1+160 #1-1312 ((#1)^2)+2480 ((#1)^3)-416 ((#1)^4)+1920 ((#1)^5)+832 ((#1)^6)-1280 ((#1)^7)+256 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-0.0275", FEExpressionsNumericalApproximationElider]), Root[1+56 #1+832 ((#1)^2)+4608 ((#1)^3)+10400 ((#1)^4)+8832 ((#1)^5)+3392 ((#1)^6)+1024 ((#1)^7)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["3.06", FEExpressionsNumericalApproximationElider]), Root[-1458650399-5312973220 #1-4264999300 ((#1)^2)+3261379040 ((#1)^3)+6856485744 ((#1)^4)+12330873920 ((#1)^5)+9901261760 ((#1)^6)-16529662720 ((#1)^7)-18399839744 ((#1)^8)+11355130880 ((#1)^9)+11007411200 ((#1)^10)-5310545920 ((#1)^11)-2778853376 ((#1)^12)+1443676160 ((#1)^13)+219545600 ((#1)^14)-186777600 ((#1)^15)+23658496 ((#1)^16)&,8], ]),(NumericalApproximation[Root, (RowDefault["0.939", FEExpressionsNumericalApproximationElider]), Root[58917025+26321800 #1-262782100 ((#1)^2)-3225600 ((#1)^3)+318373680 ((#1)^4)-467019200 ((#1)^5)-50621760 ((#1)^6)+1073716480 ((#1)^7)+48912896 ((#1)^8)-660988928 ((#1)^9)-185449472 ((#1)^10)-134209536 ((#1)^11)-9236480 ((#1)^12)+95125504 ((#1)^13)+60162048 ((#1)^14)+72220672 ((#1)^15)+23658496 ((#1)^16)&,8], ])},{(NumericalApproximation[Root, (RowDefault["3.08", FEExpressionsNumericalApproximationElider]), Root[5-10 ((#1)^2)+((#1)^4)&,4], ]),-1},{(1/4) (5+3 (Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-0.951", FEExpressionsNumericalApproximationElider]), Root[5-20 ((#1)^2)+16 ((#1)^4)&,1], ])},{(NumericalApproximation[Root, (RowDefault["3.16", FEExpressionsNumericalApproximationElider]), Root[1-8 #1-56 ((#1)^2)-32 ((#1)^3)+16 ((#1)^4)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.997", FEExpressionsNumericalApproximationElider]), Root[-79+2288 ((#1)^2)-4256 ((#1)^4)+1792 ((#1)^6)+256 ((#1)^8)&,6], ])},{(NumericalApproximation[Root, (RowDefault["2.62", FEExpressionsNumericalApproximationElider]), Root[895-94480 #1+165620 ((#1)^2)-2400 ((#1)^3)-194416 ((#1)^4)+96576 ((#1)^5)+35264 ((#1)^6)-30464 ((#1)^7)+4864 ((#1)^8)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.84", FEExpressionsNumericalApproximationElider]), Root[1-180120 ((#1)^2)+268337264 ((#1)^4)-59744000 ((#1)^6)+530759936 ((#1)^8)-249978880 ((#1)^10)-300814336 ((#1)^12)+17203200 ((#1)^14)+23658496 ((#1)^16)&,8], ])},{1+(Sqrt[5]),0},{(NumericalApproximation[Root, (RowDefault["3.08", FEExpressionsNumericalApproximationElider]), Root[5-10 ((#1)^2)+((#1)^4)&,4], ]),0}};

g=Graph[Range[Length[v]],UnitEdges[v],VertexCoordinates->v]

RecognizeGraph[g]

Non-snarks

WrongFlowerSnark1

From Read and Wilson, p. 276 "Flower snark (Subscript[J, 5])"

WrongFlowerSnark1[n_]:=Module[{e=
Join[
Partition[Range[n],2,1,1],
{#,#+n}&/@Range[n],
Partition[Range[2n+1,4n],2,1,1],
{#,2#}&/@Range[n+1,2n],
{#,2#-3}&/@Range[n+2,2n],
{{n+1,4n-1}}
],
v=Join[
CirclePoints[{3/4,Pi/n-Pi/2},n],
CirclePoints[{5/4,Pi/n-Pi/2},n],
CirclePoints[{2,π (1/2)/n+Pi/n-Pi/2},2n]]
},
Graph[Range[Length[v]],UndirectedEdge@@@Sort[Sort/@e],VertexCoordinates->v]
]

gs=Table[WrongFlowerSnark1[n],{n,3,8}]

VertexCount/@gs

CubicGraphQ/@gs

SnarkQ/@gs

gn=RecognizeGraph/@gs

TextGrid[{#,Annotate[#,ImageSize->Tiny]&/@GraphData[#,"Graph","All"]}&/@DeleteCases[gn,{}],Dividers->All]

GraphDataString[WrongFlowerSnark1[5],
"Name"->"20-cubic graph 2",
"StandardName"->{"Cubic",{20,2}},
"AlternateStandardNames"->{"FlowerNonSnarkJ5"},
Debug->True,
TimeConstraint->300
]

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

WriteString[file,GraphDataString[WrongFlowerSnark1[6],
"Name"->"24-cubic graph 4",
"StandardName"->{"Cubic",{24,4}},
Debug->True,
TimeConstraint->60
]]

WriteString[file,GraphDataString[WrongFlowerSnark1[7],
"Name"->"28-cubic graph 5",
"StandardName"->{"Cubic",{24,5}},
Debug->True,
TimeConstraint->60
]]

WriteString[file,GraphDataString[WrongFlowerSnark1[8],
"Name"->"32-cubic graph 5",
"StandardName"->{"Cubic",{32,5}},
Debug->True,
TimeConstraint->60
]]

Close[file]

{"Flower", 10}; renamed from {"Cubic", {40, 1}} in V13.1 (non-snark from Flower snark construction for (Subscript[J, 10]))

FlowerNonSnark40=Graph[Range[40],UndirectedEdge@@@{{23,17},{35,30},{39,37},{38,40},{31,36},{18,24},{6,11},{2,4},{3,1},{10,5},{23,37},{35,40},{39,36},{38,24},{31,11},{18,4},{6,1},{2,5},{3,17},{10,30},{37,32},{32,35},{27,32},{40,34},{34,39},{29,34},{36,33},{33,38},{28,33},{24,26},{26,31},{22,26},{11,16},{16,18},{20,16},{4,9},{9,6},{14,9},{1,7},{7,2},{12,7},{5,8},{8,3},{13,8},{17,15},{15,10},{19,15},{30,25},{25,23},{21,25},{21,27},{27,29},{29,28},{28,22},{22,20},{20,14},{14,12},{12,13},{13,19},{19,21}},VertexCoordinates->{{0.2113235294117647`,-0.2826105882352941`},{0.2113235294117647`,-0.17069882352941174`},{0.24959588235294114`,-0.4004782352941176`},{0.24959588235294114`,-0.05283088235294117`},{0.3154047058823529`,-0.4909929411764705`},{0.3154047058823529`,0.03768382352941176`},{0.32187529411764704`,-0.22665470588235292`},{0.3719488235294117`,-0.380735294117647`},{0.3719488235294117`,-0.0725735294117647`},{0.41566176470588234`,-0.5638235294117646`},{0.41566176470588234`,0.11051470588235293`},{0.4458088235294117`,-0.22665470588235292`},{0.47220588235294114`,-0.3079047058823529`},{0.47220588235294114`,-0.1454047058823529`},{0.5030517647058823`,-0.4759558823529411`},{0.5030517647058823`,0.02264705882352941`},{0.5220958823529411`,-0.5984176470588235`},{0.5220958823529411`,0.1451105882352941`},{0.5413235294117646`,-0.35812529411764704`},{0.5413235294117646`,-0.09518411764705881`},{0.6267647058823529`,-0.35812529411764704`},{0.6267647058823529`,-0.09518411764705881`},{0.6459941176470588`,-0.5984176470588235`},{0.6459941176470588`,0.1451105882352941`},{0.665035294117647`,-0.4759558823529411`},{0.665035294117647`,0.02264705882352941`},{0.6958823529411765`,-0.3079047058823529`},{0.6958823529411765`,-0.1454047058823529`},{0.7222823529411765`,-0.22665470588235292`},{0.7524294117647059`,-0.5638235294117646`},{0.7524294117647059`,0.11051470588235293`},{0.7961411764705882`,-0.380735294117647`},{0.7961411764705882`,-0.0725735294117647`},{0.8462117647058822`,-0.22665470588235292`},{0.8526823529411763`,-0.4909929411764705`},{0.8526823529411763`,0.03768382352941176`},{0.9184941176470588`,-0.4004782352941176`},{0.9184941176470588`,-0.05283088235294117`},{0.9567647058823529`,-0.2826105882352941`},{0.9567647058823529`,-0.17069882352941174`}}]

RecognizeGraph[FlowerNonSnark40]

GraphData[{"Flower",10},"Graph","All"]

WrongFlowerSnark2 = {"RegularNonplanarDiameter", {3, 3, 20}}

WrongFlowerSnark2[n_,opts___?OptionQ]:=Module[{e=
Join[
Partition[Range[n],2,1,1],
{#,#+n}&/@Range[n],
Partition[Range[2n+1,4n],2,1,1],
{#,2#+1}&/@Range[n+1,2n-1],
{{10,11},{6,18},{7,20},{8,12},{9,14},{10,16}}
],
v=Join[
CirclePoints[{3/4,Pi/n-Pi/2},n],
CirclePoints[{5/4,Pi/n-Pi/2},n],
CirclePoints[{5/2,π (1/2)/n+Pi/n-Pi/2},2n]]
},
Graph[Range[Length[v]],UndirectedEdge@@@Sort[Sort/@e],VertexCoordinates->v,opts]
]/;n==5

g=WrongFlowerSnark2[5,VertexLabels->Automatic]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

WrongFlowerSnark3 = {"Cubic", {20, 2}}

WrongFlowerSnark3[n_,opts___?OptionQ]:=Module[{e=
Join[
Partition[Range[n],2,1,1],
{#,#+n}&/@Range[n],
Partition[Range[2n+1,4n],2,1,1],
{#,2#+2}&/@Range[n+1,2n-1],
{{10,12},
{8,11},{9,13},{10,15},{7,19},{6,17}}
],
v=Join[
CirclePoints[{3/4,Pi/n-Pi/2},n],
CirclePoints[{5/4,Pi/n-Pi/2},n],
CirclePoints[{2,π (1/2)/n+Pi/n-Pi/2},2n]]
},
Graph[Range[Length[v]],UndirectedEdge@@@Sort[Sort/@e],VertexCoordinates->v,opts]
]/;n==5

g=WrongFlowerSnark3[5,VertexLabels->Automatic]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

{"UnitDistance", {9, 1}} (contracted (Subscript[J, 4]))

g9=IndexGraph@ContractCoincidentVertices[FlowerSnark[4]]

VertexCount[g9]

UnitDistanceGraphEmbeddingQ[g9]

Intersection[GraphData["UnitDistance*"],GraphData[13]]

gg=Graph[Range[13],Sort[Sort/@EdgeList[g9]],VertexCoordinates->GraphEmbedding[g9]]

gg=;

RecognizeGraph[gg]

GraphData[{"UnitDistance", {9, 1}},"Name"]//FullForm

file=OpenWrite["~/ud13.m"]

WriteString[file,GraphDataString[gg,
"Name"->"13\[Hyphen]unit distance graph 1",
"StandardName"->{"UnitDistance",{13,1}},
"Information"->"Unit-DistanceGraph",
"Skip"->{"MinimumDistinguishingLabelings"},
Debug->True,
TimeConstraint->10]]

Close[file]

Snark status

TextGrid[Prepend[
Prepend[GraphData[#,{"V","Snark","#","Cubic","Class2","EdgeChromaticNumber"}],GraphData[#,"NotationRules","Flower"]]&/@GraphData["Flower"],
Thread[Style[{"n","V","snark","name","cubic","class 2","ECN"},Bold]]
],Dividers->All,Alignment->{{Decimal,Decimal,{Left}}}]

Properties

Possibly (Subscript[J, 6]) should be defined differently?

Table[GraphData[{"Flower",n},"Regular"],{n,k0=5,20}]

Table[GraphData[{"Flower",n},"EdgeCount"],{n,k0=5,20}]

Although:

Table[GraphData[{"Flower",n},"VertexCount"],{n,k0=5,20}]

Acyclic

(seq=Table[GraphData[{"Flower",n},"Acyclic"],{n,5,20}])//Union

AdjacencyMatrixCount

seq=Table[GraphData[{"Flower",n},"AdjacencyMatrixCount"],{n,5,20}]

AlgebraicConnectivity

seq=Table[GraphData[{"Flower",n},"AlgebraicConnectivity"],{n,5,20}]

AlmostHamiltonian

seq=Table[GraphData[{"Flower",n},"AlmostHamiltonian"],{n,5,20}]

Anarboricity

seq=Table[GraphData[{"Flower",n},"Anarboricity"],{n,5,20}]

Antipodal

Table[GraphData[{"Flower",n},"Antipodal"],{n,5,20}]//Union

Apex

seq=Table[GraphData[{"Flower",n},"Apex"],{n,5,20}]

ApexCount

seq=Table[GraphData[{"Flower",n},"ApexCount"],{n,5,20}]

Arboricity

Table[GraphData[{"Flower",n},"Arboricity"],{n,5,20}]

ArcTransitive

Table[GraphData[{"Flower",n},"ArcTransitive"],{n,5,20}]//Union

ArcTransitivity

Table[GraphData[{"Flower",n},"ArcTransitivity"],{n,5,20}]

ArticulationVertexCount

Table[GraphData[{"Flower",n},"ArticulationVertexCount"],{n,5,20}]

AutomorphismCount

seq=Table[GraphData[{"Flower",n},"AutomorphismCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]

gf=FindGeneratingFunction[(x^5)seq,x]

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

BalabanIndex

seq=Table[GraphData[{"Flower",n},"BalabanIndex"],{n,5,20}]

Bandwidth

Table[GraphData[{"Flower",n},"Bandwidth"],{n,5,20}]

BetweennessCentralities

Table[GraphData[{"Flower",n},"BetweennessCentralities"],{n,5,20}]

Bicolorable

4+Position[Table[GraphData[{"Flower",n},"Bicolorable"],{n,5,20}],True]//Flatten

Biconnected

Table[GraphData[{"Flower",n},"Biconnected"],{n,5,20}]//Union

Bicubic

4+Position[Table[GraphData[{"Flower",n},"Bicubic"],{n,5,20}],True]//Flatten

Bipartite

4+Position[Table[GraphData[{"Flower",n},"Bipartite"],{n,5,20}],True]//Flatten

BipartiteDimension

Table[GraphData[{"Flower",n},"BipartiteDimension"],{n,5,20}]

Biplanar

4+Position[Table[GraphData[{"Flower",n},"Biplanar"],{n,5,20}],True]//Flatten

BlockCount

seq=Table[GraphData[{"Flower",n},"BlockCount"],{n,5,20}]

BridgeCount

seq=Table[GraphData[{"Flower",n},"BridgeCount"],{n,5,20}]

Bridged

Table[GraphData[{"Flower",n},"Bridged"],{n,5,20}]//Union

BroadcastTime

seq=Table[GraphData[{"Flower",n},"BroadcastTime"],{n,5,20}]

BurningNumber

seq=Table[GraphData[{"Flower",n},"BurningNumber"],{n,5,20}]

CharacteristicPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"CharacteristicPolynomial"][x],{n,k0=5,20}])//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[(((-1-x))^(-1+n)) (-1+n-x),{n,k0,20}]

FullSimplify[poly2-poly]

CheegerConstant

Chordal

ChordCount

Computed up to n==17; inferred beyond

seq=Table[GraphData[{"Flower",n},"ChordCount"],{n,5,20}]

gf=(x^5) FindGeneratingFunction[DeleteMissing@seq,x]

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

ChordlessCycleCount

seq=Table[GraphData[{"Flower",n},"ChordlessCycleCount"],{n,5,20}]

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

ChromaticInvariant

seq=Table[GraphData[{"Flower",n},"ChromaticInvariant"],{n,5,20}]

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

ChromaticNumber

seq=Table[GraphData[{"Flower",n},"ChromaticNumber"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>6]

ChromaticPolynomial

seq=Table[GraphData[{"Flower",n},"ChromaticPolynomial"],{n,5,20}]/.f_Function:>f[x]

FindSequenceFunction[DeleteMissing[seq],n+4]

FindGeneratingFunction[DeleteMissing@seq,z]//Head//Timing

CircuitRank

seq=Table[GraphData[{"Flower",n},"CircuitRank"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

Circumference

seq=Table[GraphData[{"Flower",n},"Circumference"],{n,5,20}]

FindSequenceFunction[seq,n-4]//FullSimplify

Classes

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

ClawFree

Table[GraphData[{"Flower",n},"ClawFree"],{n,5,20}]//Union

CliqueCount

seq=Table[GraphData[{"Flower",n},"CliqueCount"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

CliqueCoveringNumber

seq=Table[GraphData[{"Flower",n},"CliqueCoveringNumber"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

CliqueNumber

seq=Table[GraphData[{"Flower",n},"CliqueNumber"],{n,5,20}]

CliquePolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"CliquePolynomial"][x],{n,k0=5,20}])//Column

gf=(z^5)FindGeneratingFunction[poly,z]

SeriesCoefficient[gf,{z,0,n}]//Factor

ClosenessCentralities

Tally/@Table[GraphData[{"Flower",n},"ClosenessCentralities"],{n,5,20}]

Coarseness

seq=Table[GraphData[{"Flower",n},"Coarseness"],{n,5,20}]

CoboundaryPolynomial

ComplementOddChordlessCycleCount

seq=Table[GraphData[{"Flower",n},"ComplementOddChordlessCycleCount"],{n,5,20}]

ComplementOddChordlessCyclePolynomial

Conductance

Connected

ConnectedComponentCount

ConnectedDominatingSetCount

seq=Table[GraphData[{"Flower",n},"ConnectedDominatingSetCount"],{n,5,20}]

ConnectedDominationNumber

seq=Table[GraphData[{"Flower",n},"ConnectedDominationNumber"],{n,5,20}]

ConnectedDominationPolynomial

(poly=DeleteMissing[Factor@Table[GraphData[{"Flower",n},"ConnectedDominationPolynomial"],{n,k0=5,20}]]/.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,5,20}]

FullSimplify[poly2-poly]

ConnectedInducedSubgraphCount

seq=Table[GraphData[{"Flower",n},"ConnectedInducedSubgraphCount"],{n,5,20}]

ConnectedInducedSubgraphPolynomial

(poly=DeleteMissing[Factor@Table[GraphData[{"Flower",n},"ConnectedInducedSubgraphPolynomial"],{n,k0=5,20}]]/.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,5,20}]

FullSimplify[poly2-poly]

Corank

seq=Table[GraphData[{"Flower",n},"Corank"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

CriticalNonplanar

CrossingNumber

seq=Table[GraphData[{"Flower",n},"CrossingNumber"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

Cubic

4+Position[Table[GraphData[{"Flower",n},"Cubic"],{n,5,20}],False]//Flatten

CycleCount

seq=Table[GraphData[{"Flower",n},"CycleCount"],{n,5,20}]

Do[Print[{k,FindLinearRecurrence[Drop[DeleteMissing[seq],k]]//Head//Timing}],{k,0,4}]

FindSequenceFunction[DeleteMissing[seq],n-4]//Head//Timing

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

CyclePolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"CyclePolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

FindGeneratingFunction[DeleteMissing@poly,z]//Head//Timing

CoefficientList[poly,x]

Cyclic

Table[GraphData[{"Flower",n},"Cyclic"],{n,5,20}]//Union

Degeneracy

Table[GraphData[{"Flower",n},"Degeneracy"],{n,5,20}]

DegreeCentralities

Tally/@Table[GraphData[{"Flower",n},"DegreeCentralities"],{n,5,20}]

DetourIndex

seq=Table[GraphData[{"Flower",n},"DetourIndex"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n+4]//Head//Timing

DetourPolynomial

(poly=Factor@DeleteMissing[Table[GraphData[{"Flower",n},"DetourPolynomial"],{n,k0=5,20}]/.f_Function:>f[x]])//Column

coef=Factor/@FindLinearRecurrence[poly]

poly2=Table[(((n-1)^2-#) (#+n-1)^(n-1)&)@x,{n,k0,20}]

Abs[FullSimplify[poly2/poly]]

Diameter

seq=Table[GraphData[{"Flower",n},"Diameter"],{n,5,20}]

Dimension

seq=Table[GraphData[{"Flower",n},"Dimension"],{n,5,20}]

DistancePolynomial

(poly=DeleteMissing[Factor@Table[GraphData[{"Flower",n},"DistancePolynomial"],{n,k0=5,20}]]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@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[DeleteMissing@poly,z]]

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

Table[n x(3x+2)-2x^2+1,{n,10}]

FullSimplify[%-poly]

DistanceRegular

Table[GraphData[{"Flower",n},"DistanceRegular"],{n,5,20}]//Union

DistanceTransitive

Table[GraphData[{"Flower",n},"DistanceTransitive"],{n,5,20}]//Union

DistinguishingNumber

seq=Table[GraphData[{"Flower",n},"DistinguishingNumber"],{n,5,20}]

DomaticNumber

seq=Table[GraphData[{"Flower",n},"DomaticNumber"],{n,5,20}]

DominatingSetCount

seq=Table[GraphData[{"Flower",n},"DominatingSetCount"],{n,5,20}]

FindGeneratingFunction[DeleteMissing@seq,x]

DominationNumber

seq=Table[GraphData[{"Flower",n},"DominationNumber"],{n,5,20}]

FindSequenceFunction[seq,n-4]

DominationPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"DominationPolynomial"],{n,k0=5,20}]/.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[-1+(((1+x))^n),{n,k0,20}]

FullSimplify[poly2-poly]

Doublecross

Table[GraphData[{"Flower",n},"Doublecross"],{n,5,20}]//Union

Eccentricities

Tally/@Table[GraphData[{"Flower",n},"Eccentricities"],{n,5,20}]

EccentricityCentralities

Tally/@Table[GraphData[{"Flower",n},"EccentricityCentralities"],{n,5,20}]

EdgeBetweennessCentralities

Table[GraphData[{"Flower",n},"EdgeBetweennessCentralities"],{n,5,20}]

EdgeChromaticNumber

seq=Table[GraphData[{"Flower",n},"EdgeChromaticNumber"],{n,5,20}]

EdgeConnectivity

Table[GraphData[{"Flower",n},"EdgeConnectivity"],{n,5,20}]

EdgeCount

seq=Table[GraphData[{"Flower",n},"EdgeCount"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

EdgeCoverCount

seq=Table[GraphData[{"Flower",n},"EdgeCoverCount"],{n,5,20}]

EdgeCoverNumber

seq=Table[GraphData[{"Flower",n},"EdgeCoverNumber"],{n,5,20}]

FindSequenceFunction[seq,n+4]

EdgeCoverPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"EdgeCoverPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

Sum[(-1)^(n-\[FormalK]) Binomial[n,\[FormalK]] (x+1)^Binomial[\[FormalK],2],{\[FormalK],0,n}]

poly2=Table[Sum[(-1)^(n-\[FormalK]) Binomial[n,\[FormalK]] (x+1)^Binomial[\[FormalK],2],{\[FormalK],0,n}],{n,k0,20}]

FullSimplify[Take[poly2-poly,7]]

EdgeTransitive

Table[GraphData[{"Flower",n},"EdgeTransitive"],{n,5,20}]//Union

EigenvectorCentralities

Tally/@Table[GraphData[{"Flower",n},"EigenvectorCentralities"],{n,5,20}]

EulerianCycleCount

seq=Table[GraphData[{"Flower",n},"EulerianCycleCount"],{n,5,20}]

FaceCount

seq=Table[GraphData[{"Flower",n},"FaceCount"],{n,5,20}]

FlowPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"FlowPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[poly]

Take[poly2=Table[Block[{F},F[0,{_,_}]:=1;
F[1,{x_,_}]:=x;
F[nn_,{x_,y_}]:=F[nn,{x,y}]=Factor@Sum[Binomial[nn-1,\[FormalK]-1] (x+(y^\[FormalK]-y)/(y-1)) F[\[FormalK]-1,{1,y}] F[nn-\[FormalK],{x,y}],{\[FormalK],nn}];
Function[u,Evaluate[(-1)^Binomial[n-1,2] F[n-1,{0,1-u}]]]][x],{n,k0,20}],UpTo[6]]

FullSimplify[poly2-poly]

FractionalArboricity

seq=Table[GraphData[{"Flower",n},"FractionalArboricity"],{n,5,20}]

FractionalChromaticNumber

seq=Table[GraphData[{"Flower",n},"FractionalChromaticNumber"],{n,5,20}]

Table[Piecewise[{{2,Mod[n,2]==0}},((2n)/(n-1))],{n,5,20}]

FractionalEdgeChromaticNumber

seq=Table[GraphData[{"Flower",n},"FractionalEdgeChromaticNumber"],{n,5,20}]

Genus

seq=Table[GraphData[{"Flower",n},"Genus"],{n,5,20}]

Geodetic

Table[GraphData[{"Flower",n},"Geodetic"],{n,5,20}]//Union

Girth

seq=Table[GraphData[{"Flower",n},"Girth"],{n,5,20}]

GlobalClusteringCoefficient

seq=Table[GraphData[{"Flower",n},"GlobalClusteringCoefficient"],{n,5,20}]

Gonality

seq=Table[GraphData[{"Flower",n},"Gonality"],{n,5,20}]

Graceful

Table[GraphData[{"Flower",n},"Graceful"],{n,5,20}]

GracefulLabelingCount

seq=Table[GraphData[{"Flower",n},"GracefulLabelingCount"],{n,5,20}]

HamiltonConnected

Table[GraphData[{"Flower",n},"HamiltonConnected"],{n,5,20}]

HamiltonDecomposable

4+Position[Table[GraphData[{"Flower",n},"HamiltonDecomposable"],{n,5,20}],True]//Flatten

HamiltonDecompositionCount

seq=Table[GraphData[{"Flower",n},"HamiltonDecompositionCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]

Hamiltonian

4+Position[Table[GraphData[{"Flower",n},"Hamiltonian"],{n,5,20}],True]//Flatten

HamiltonianCycleCount

seq=Table[GraphData[{"Flower",n},"HamiltonianCycleCount"],{n,5,20}]

seq[[4;;All;;2]]

FindSequenceFunction[%,n/2-2]//FullSimplify

FullSimplify[-4 (I^(n-2))+(2^(1+(n/2))),Mod[n,2]==0]

Table[Piecewise[{{2 (2 (((-1))^(n/2))+(2^(n/2))),Mod[n,2]==0}}],{n,5,20}]

HamiltonianNumber

seq=Table[GraphData[{"Flower",n},"HamiltonianNumber"],{n,5,20}]

HamiltonianPathCount

seq=Table[GraphData[{"Flower",n},"HamiltonianPathCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing[seq],n-4]//Head//Timing

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

HamiltonianWalkCount

seq=Table[GraphData[{"Flower",n},"HamiltonianWalkCount"],{n,5,20}]

HamiltonLaceable

Table[GraphData[{"Flower",n},"HamiltonLaceable"],{n,5,20}]

HararyIndex

seq=Table[GraphData[{"Flower",n},"HararyIndex"],{n,5,20}]

FindLinearRecurrence[seq]//Head//Timing

FindSequenceFunction[seq,n-4]//Head//Timing

gf=(x^5)FindGeneratingFunction[seq,x]

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

HexagonCount

seq=Table[GraphData[{"Flower",n},"HexagonCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]

gf=(x^5)FindGeneratingFunction[seq,x]

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

HosoyaIndex

seq=Table[GraphData[{"Flower",n},"HosoyaIndex"],{n,5,20}]

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

HStarConnected

Table[GraphData[{"Flower",n},"HStarConnected"],{n,5,20}]

Hypohamiltonian

Table[GraphData[{"Flower",n},"HStarConnected"],{n,5,20}]

Hypotraceable

Table[GraphData[{"Flower",n},"HStarConnected"],{n,5,20}]

Identity

Table[GraphData[{"Flower",n},"Identity"],{n,5,20}]//Union

IdiosyncraticPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"IdiosyncraticPolynomial"][x,y],{n,k0=5,20}])//Column

FindLinearRecurrence[poly]//Head//Timing

FindGeneratingFunction[poly,z]//Head//Timing

IndependenceNumber

seq=Table[GraphData[{"Flower",n},"IndependenceNumber"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

IndependencePolynomial

(poly=DeleteMissing[Factor@Table[GraphData[{"Flower",n},"IndependencePolynomial"],{n,k0=5,20}]/.f_Function:>f[x]])//Column

FindLinearRecurrence[poly]//Head//Timing

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]

FindGeneratingFunction[poly,z]//Head//Timing

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

poly2=Table[n x+1,{n,l0=10}]

FullSimplify[poly2-poly]

IndependenceRatio

seq=Table[GraphData[{"Flower",n},"IndependenceRatio"],{n,5,20}]

FindSequenceFunction[seq,n-4]

seq[[4;;All;;2]]

seq[[5;;All;;2]]

FindSequenceFunction[%,n]

Table[((13+4 n)/(4 (7+2 n))),{n,5}]

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

IndependentEdgeSetCount

seq=Table[GraphData[{"Flower",n},"IndependentEdgeSetCount"],{n,5,20}]

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

IndependentVertexSetCount

seq=Table[GraphData[{"Flower",n},"IndependentVertexSetCount"],{n,5,20}]

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

IntersectionNumber

seq=Table[GraphData[{"Flower",n},"IntersectionNumber"],{n,5,20}]

IrredundanceNumber

seq=Table[GraphData[{"Flower",n},"IrredundanceNumber"],{n,5,20}]

IrredundancePolynomial

(poly=DeleteMissing[Table[GraphData[{"Flower",n},"IrredundancePolynomial"],{n,5,20}]]/.f_Function:>f[x])

IrredundantSetCount

seq=Table[GraphData[{"Flower",n},"IrredundantSetCount"],{n,5,20}]

IsolatedPointCount

seq=Table[GraphData[{"Flower",n},"IsolatedPointCount"],{n,5,20}]

KirchhoffIndex

seq=Table[GraphData[{"Flower",n},"KirchhoffIndex"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

KirchhoffSumIndex

seq=Table[GraphData[{"Flower",n},"KirchhoffSumIndex"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

LaplacianPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"LaplacianPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

FindGeneratingFunction[poly,z]//Head//Timing

LeafCount

seq=Table[GraphData[{"Flower",n},"LeafCount"],{n,5,20}]

Likelihood

seq=Table[GraphData[{"Flower",n},"Likelihood"],{n,5,20}]

LinearArboricity

seq=Table[GraphData[{"Flower",n},"LinearArboricity"],{n,5,20}]

LocalClusteringCoefficients

Tally/@Table[GraphData[{"Flower",n},"LocalClusteringCoefficients"],{n,5,20}]//Column

LongestCycleCount

seq=Table[GraphData[{"Flower",n},"LongestCycleCount"],{n,5,20}]

LongestPathCount

seq=Table[GraphData[{"Flower",n},"LongestPathCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n+4]//Head//Timing

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

LongestPathLength

seq=Table[GraphData[{"Flower",n},"LongestPathLength"],{n,5,20}]

FindGeneratingFunction[seq,n+4]//Factor

LovaszNumber

seq=Table[GraphData[{"Flower",n},"LovaszNumber"],{n,5,20}]

LowerIndependenceNumber

Computed for n<=14; inferred beyond

seq=Table[GraphData[{"Flower",n},"LowerIndependenceNumber"],{n,5,20}]

LowerMatchingNumber

A004772

Computed for n<=14; inferred beyond

seq=Table[GraphData[{"Flower",n},"LowerMatchingNumber"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[DeleteMissing@seq,x]

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

Table[Ceiling[((4n)/3)],{n,5,20}]

MatchingGeneratingPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"MatchingGeneratingPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

Do[Print[{k,FindLinearRecurrence[Drop[DeleteMissing@poly,k]]//Head//Timing}],{k,0,6}]

Do[Print[{k,FindGeneratingFunction[Drop[DeleteMissing@poly,k],z]//Head//Timing}],{k,0,6}]

MatchingNumber

seq=Table[GraphData[{"Flower",n},"MatchingNumber"],{n,5,20}]

MatchingPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"MatchingPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

MaximalCliqueCount

seq=Table[GraphData[{"Flower",n},"MaximalCliqueCount"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

MaximalCliquePolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"MaximalCliquePolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

MaximalIndependencePolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"MaximalIndependencePolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

FindLinearRecurrence[Drop[DeleteMissing@poly],2]//Head//Timing

FindGeneratingFunction[DeleteMissing@poly,z]//Head//Timing

MaximalIndependentEdgeSetCount

seq=Table[GraphData[{"Flower",n},"MaximalIndependentEdgeSetCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing[seq],n+4]//Head//Timing

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

MaximalIndependentVertexSetCount

seq=Table[GraphData[{"Flower",n},"MaximalIndependentVertexSetCount"],{n,5,20}]

4+Position[seq,_Missing]//Flatten

FindSequenceFunction[DeleteMissing[seq],n+4]//Head//Timing

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

MaximalIrredundancePolynomial

MaximalIrredundantSetCount

seq=Table[GraphData[{"Flower",n},"MaximalIrredundantSetCount"],{n,5,20}]

MaximalMatchingGeneratingPolynomial

MaximumCliqueCount

seq=Table[GraphData[{"Flower",n},"MaximumCliqueCount"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

MaximumIndependentEdgeSetCount

seq=Table[GraphData[{"Flower",n},"MaximumIndependentEdgeSetCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

MaximumIndependentVertexSetCount

seq=Table[GraphData[{"Flower",n},"MaximumIndependentVertexSetCount"],{n,5,20}]

MaximumIrredundantSetCount

seq=Table[GraphData[{"Flower",n},"MaximumLeafNumber"],{n,5,20}]

MaximumLeafNumber

seq=Table[GraphData[{"Flower",n},"MaximumLeafNumber"],{n,5,20}]

MaximumVertexDegree

seq=Table[GraphData[{"Flower",n},"MaximumVertexDegree"],{n,5,20}]

MeanClusteringCoefficient

seq=Table[GraphData[{"Flower",n},"MeanClusteringCoefficient"],{n,5,20}]

MeanDistance

seq=Table[GraphData[{"Flower",n},"MeanDistance"],{n,5,20}]

FindSequenceFunction[seq[[1;;All;;2]],n]//Factor

FindSequenceFunction[seq[[4;;All;;2]],n]//Factor

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

Median

Table[GraphData[{"Flower",n},"Median"],{n,5,20}]//Union

MetricDimension

MinimalConnectedDominatingSetCount

MinimalConnectedDominationPolynomial

MinimalDominatingSetCount

seq=Table[GraphData[{"Flower",n},"MinimalDominatingSetCount"],{n,5,20}]

MinimalDominatingSetSignature

seq=Table[GraphData[{"Flower",n},"MinimalDominatingSetSignature"],{n,5,20}]

MinimalDominationPolynomial

MinimalEdgeCoverCount

seq=Table[GraphData[{"Flower",n},"MinimalEdgeCoverCount"],{n,5,20}]

MinimalEdgeCoverPolynomial

MinimalTotalDominatingSetCount

seq=Table[GraphData[{"Flower",n},"MinimalTotalDominatingSetCount"],{n,5,20}]

MinimalTotalDominationPolynomial

MinimalVertexCoverCount

seq=Table[GraphData[{"Flower",n},"MinimalVertexCoverCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n +4]//Head//Timing

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

MinimalVertexCoverPolynomial

MinimumCliqueCoveringCount

seq=Table[GraphData[{"Flower",n},"MinimumCliqueCoveringCount"],{n,5,20}]

MinimumConnectedDominatingSetCount

seq=Table[GraphData[{"Flower",n},"MinimumConnectedDominatingSetCount"],{n,5,20}]

FindSequenceFunction[DeleteCases[seq,_Missing],n]

MinimumCoveringsByMaximalCliquesCount

seq=Table[GraphData[{"Flower",n},"MinimumCoveringsByMaximalCliquesCount"],{n,5,20}]

MinimumDistinguishingLabelingCount

seq=Table[GraphData[{"Flower",n},"MinimumDistinguishingLabelingCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]

MinimumDominatingSetCount

seq=Table[GraphData[{"Flower",n},"MinimumDominatingSetCount"],{n,5,20}]

MinimumEdgeCoverCount

seq=Table[GraphData[{"Flower",n},"MinimumEdgeCoverCount"],{n,5,20}]

MinimumLeafNumber

seq=Table[GraphData[{"Flower",n},"MinimumLeafNumber"],{n,5,20}]

MinimumPathCoveringCount

seq=Table[GraphData[{"Flower",n},"MinimumPathCoveringCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n+4]//Head//Timing

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

MinimumTotalDominatingSetCount

seq=Table[GraphData[{"Flower",n},"MinimumTotalDominatingSetCount"],{n,5,20}]

MinimumVertexCoverCount

seq=Table[GraphData[{"Flower",n},"MinimumVertexCoverCount"],{n,5,20}]

MinimumVertexCutCount

seq=Table[GraphData[{"Flower",n},"MinimumVertexCutCount"],{n,5,20}]

MinimumVertexDegree

seq=Table[GraphData[{"Flower",n},"MinimumVertexDegree"],{n,5,20}]

MolecularTopologicalIndex

seq=Table[GraphData[{"Flower",n},"MolecularTopologicalIndex"],{n,5,20}]

FindSequenceFunction[seq,n-4]//FullSimplify

gf=(x^5)FindGeneratingFunction[seq,x]

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

NoPerfectMatching

OddChordlessCycleCount

seq=Table[GraphData[{"Flower",n},"OddChordlessCycleCount"],{n,5,20}]

OddChordlessCyclePolynomial

Outerplanar

PageRankCentralities

PathCount

seq=Table[GraphData[{"Flower",n},"PathCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n-4]//Head

FindLinearRecurrence[DeleteMissing@seq]//Head//Timing

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

PathCoveringNumber

seq=Table[GraphData[{"Flower",n},"PathCoveringNumber"],{n,5,20}]

PathPolynomial

(poly=Factor[Table[GraphData[{"Flower",n},"PathPolynomial"],{n,k0=5,kmax=20}]/.f_Function:>f[x]])//Column

CoefficientList[Take[poly,4],x]

FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

FindGeneratingFunction[DeleteMissing@poly,z]//Head//Timing

PentagonCount

seq=Table[GraphData[{"Flower",n},"PentagonCount"],{n,5,20}]

Perfect

seq=Table[GraphData[{"Flower",n},"Perfect"],{n,5,20}]

4+Position[seq,True]//Flatten

PerfectMatching

seq=Table[GraphData[{"Flower",n},"PerfectMatching"],{n,5,20}]//Union

Periphery

Planar

Table[GraphData[{"Flower",n},"Planar"],{n,5,20}]//Union

ProjectivePlaneCrossingNumber

seq=Table[GraphData[{"Flower",n},"ProjectivePlaneCrossingNumber"],{n,5,20}]

Pseudoarboricity

seq=Table[GraphData[{"Flower",n},"Pseudoarboricity"],{n,5,20}]

QChromaticPolynomial

RadialityCentralities

RadioLabelingCount

seq=Table[GraphData[{"Flower",n},"RadioLabelingCount"],{n,5,20}]

RadioNumber

seq=Table[GraphData[{"Flower",n},"RadioNumber"],{n,5,20}]

Radius

seq=Table[GraphData[{"Flower",n},"Radius"],{n,5,20}]

FindSequenceFunction[seq,n-4]

gf=(x^5)FindGeneratingFunction[seq,x]

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

Rank

seq=Table[GraphData[{"Flower",n},"Rank"],{n,5,20}]

RankPolynomial

Take[poly=Factor@Table[GraphData[{"Flower",n},"RankPolynomial"][x,y],{n,k0=5,20}],4]//Column

coef=Factor/@FindLinearRecurrence[poly]

RectilinearCrossingNumber

seq=Table[GraphData[{"Flower",n},"RectilinearCrossingNumber"],{n,5,20}]

Regular

Table[GraphData[{"Flower",n},"Regular"],{n,k0=5,20}]

VertexDegree[%]

RegularParameters

ReliabilityPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"ReliabilityPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

ResistanceMatrix

MatrixForm/@Table[GraphData[{"Flower",n},"ResistanceMatrix"],{n,5,7}]

ScrambleNumber

seq=Table[GraphData[{"Flower",n},"ScrambleNumber"],{n,5,20}]

ShannonCapacity

seq=Table[GraphData[{"Flower",n},"ShannonCapacity"],{n,5,20}]

SigmaPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"SigmaPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

coef=Factor/@FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

coef=Factor/@FindPolynomialRecurrence[poly,6]

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]

Table[n x(3x+2)-2x^2+1,{n,10}]

FullSimplify[%-poly]

Singlecross

Table[GraphData[{"Flower",n},"Singlecross"],{n,5,20}]//Union

Skewness

seq=Table[GraphData[{"Flower",n},"Skewness"],{n,5,20}]

SpanningTreeCount

seq=Table[GraphData[{"Flower",n},"SpanningTreeCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

SpectralRadius

seq=Table[GraphData[{"Flower",n},"SpectralRadius"],{n,5,20}]

SquareCount

seq=Table[GraphData[{"Flower",n},"SquareCount"],{n,5,20}]

SquareFree

4+Position[Table[GraphData[{"Flower",n},"SquareFree"],{n,5,20}],False]//Flatten

StabilityIndex

seq=Table[GraphData[{"Flower",n},"StabilityIndex"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

StarArboricity

seq=Table[GraphData[{"Flower",n},"StarArboricity"],{n,5,20}]

StatusCentralities

Strength

seq=Table[GraphData[{"Flower",n},"Strength"],{n,5,20}]

StronglyPerfect

StronglyRegular

Symmetric

SymmetricallyDistinctVertexPairCount

SymmetricallyDistinctVertexPairSignature

Thickness

TopologicalIndex

seq=Table[GraphData[{"Flower",n},"TopologicalIndex"],{n,5,20}]

FindSequenceFunction[seq,n-4]//Head//Timing

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

ToroidalCrossingNumber

seq=Table[GraphData[{"Flower",n},"ToroidalCrossingNumber"],{n,5,20}]

TotalDominatingSetCount

seq=Table[GraphData[{"Flower",n},"TotalDominatingSetCount"],{n,5,20}]

TotalDominationNumber

seq=Table[GraphData[{"Flower",n},"TotalDominationNumber"],{n,5,20}]

gf=FindGeneratingFunction[seq,x]

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

TotalDominationPolynomial

Traceable

Table[GraphData[{"Flower",n},"Traceable"],{n,5,20}]//Union

TreeDepth

Table[GraphData[{"Flower",n},"TreeDepth"],{n,5,20}]

TreeNumber

Table[GraphData[{"Flower",n},"TreeNumber"],{n,5,20}]

Treewidth

Table[GraphData[{"Flower",n},"Treewidth"],{n,5,20}]

Triameter

Table[GraphData[{"Flower",n},"Triameter"],{n,5,20}]

TriangleCount

seq=Table[GraphData[{"Flower",n},"TriangleCount"],{n,5,20}]

TriangleFree

TuttePolynomial

(poly=DeleteMissing[Table[GraphData[{"Flower",n},"TuttePolynomial"],{n,5,20}]]/.f_Function:>f[x,y])//Column

Unicyclic

Table[GraphData[{"Flower",n},"Unicyclic"],{n,5,20}]//Union

UniquelyPancyclic

seq=Table[GraphData[{"Flower",n},"UniquelyPancyclic"],{n,5,20}]//Union

UnitDistance

Table[GraphData[{"Flower",n},"UnitDistance"],{n,5,20}]//Union

UpperDominationNumber

seq=Table[GraphData[{"Flower",n},"UpperDominationNumber"],{n,5,20}]

UpperIrredundanceNumber

seq=Table[GraphData[{"Flower",n},"UpperIrredundanceNumber"],{n,5,20}]

VertexConnectivity

Table[GraphData[{"Flower",n},"VertexConnectivity"],{n,5,20}]

VertexCount

seq=Table[GraphData[{"Flower",n},"VertexCount"],{n,5,20}]

FindSequenceFunction[seq,n-4]

VertexCoverCount

seq=Table[GraphData[{"Flower",n},"VertexCoverCount"],{n,5,20}]

FindSequenceFunction[DeleteMissing@seq,n-4]//Head//Timing

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

VertexCoverNumber

seq=Table[GraphData[{"Flower",n},"VertexCoverNumber"],{n,5,20}]

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

VertexCoverPolynomial

(poly=Factor@Table[GraphData[{"Flower",n},"VertexCoverPolynomial"],{n,k0=5,20}]/.f_Function:>f[x])//Column

Factor/@FindLinearRecurrence[DeleteMissing@poly]//Head//Timing

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[(x^(n-1)) (x+n),{n,10}]

FullSimplify[poly2-poly]

VertexDegrees

Tally/@Table[GraphData[{"Flower",n},"VertexDegrees"],{n,5,20}]//Column

Table[ConstantArray[n-1,n],{n,5}]==seq

VertexTransitive

Table[GraphData[{"Flower",n},"VertexTransitive"],{n,5,20}]//Union

WienerIndex

seq=Table[GraphData[{"Flower",n},"WienerIndex"],{n,5,20}]

gf=(x^5)FindGeneratingFunction[seq,x]

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

WienerSumIndex

seq=Table[GraphData[{"Flower",n},"WienerSumIndex"],{n,5,20}]

FindSequenceFunction[seq,n+4]//Head//Timing

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

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

GraphData

GraphDataString

{"Flower", n} (non-snarks for even n)

<<MathWorld`Graphs`

gs=Table[FlowerSnark[n],{n,6,10,2}]

VertexCount/@%

RecognizeGraph/@gs

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

Do[WriteString[file,GraphDataString[g=FlowerSnark[n],
"Name"->ToString[n]<>"\[Hyphen]flower graph",
"StandardName"->{"Flower",n},
"Information"->"FlowerSnark",
"Classes"->{"Flower"},
"Notation"->Subscript["J",n],
"NotationRules"->{"Flower"->n},
"Skip"->{"Arboricity","ConnectedInducedSubgraphPolynomial","EdgeChromaticNumber","MaximalIndependentVertexSets","MinimumEdgeColoring","MinimumVertexColoring"},
Debug->True,
TimeConstraint->10
]],{n,16,20,2}]

Close[file]