Ruby Graph

Author

Eric W. Weisstein
July 27, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/GraphTheory/RubyGraph.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Figure

GraphicsGrid[
Partition[
StyleGraphs[
GraphData["RubyGraph","Graph","All"]
],
UpTo[5]
],
Dividers->All
]

Embeddings

<<MathWorld`Graphs`

Names

GraphData["RubyGraph","StandardName"]

GraphData["RubyGraph","StandardNames"]

HouseOfGraphs

hogid=GraphData["RubyGraph","NotationRules","HouseOfGraphs"]

HouseOfGraphsLookup[hogid]

HouseOfGraphsGraph[hogid]//StyleGraphs[#,ImageSize->Small]&

Wikidata

GraphData["RubyGraph","WikidataID"]

GraphData["RubyGraph","WikidataID","URL"]

Robertson et al. 2019

RegionIntersection[InfiniteLine[v[[{3,4}]]],InfiniteLine[v[[{11,12}]]]]

v={{-2,0},{0,-2},{2,0},{0,2},{0,-1},{0,1},{1,1/4},{1,-1/4},{-1,1/4},{-1,-1/4}};
v=Join[v,{Midpoint[v[[{6,9}]]],Midpoint[v[[{6,7}]]],Midpoint[v[[{5,10}]]],Midpoint[v[[{5,8}]]],{(11/8),(5/8)},{(11/8),-(5/8)},{((-11)/8),(5/8)},-{(11/8),(5/8)}}];
ruby=Graph[Range[Length[v]],UndirectedEdge@@@Sort[{{2,5},{4,6},{7,8},{9,10},{11,14},{12,13},{7,12},{6,12},{6,11},{9,11},{10,13},{5,13},{5,14},{8,14},{2,16},{8,16},{3,16},{3,15},{7,15},{4,15},{4,17},{9,17},{1,17},{1,18},{10,18},{2,18},{1,3}}],VertexCoordinates->v,VertexLabels->"Name"]

v//FullSimplify

CubicGraphQ[ruby]

VertexDegree[ruby]

r=RecognizeGraph[ruby]

CrossingNumber[ruby,TimeConstraint->10]

AlmostPlanarGraphQ[ruby]

Primary

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

All

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

CurvedEdge

GraphData["RubyGraph","Graph","CurvedEdge"]//StyleGraphs

Inexact

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

Degenerate

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

DegenerateGraphEmbeddingTypes/@GraphData["RubyGraph","Graph","Degenerate"]//Column

Embeddings

GraphDataEmbeddings[GraphData["RubyGraph"]]

Default embedding

g0=GraphPlot[Graph[GraphData["RubyGraph","VertexList"],GraphData["RubyGraph","Edges"]]]

g=ToGridEmbedding@

CirclePoints[1/4,2]

v=Join[CirclePoints[{1,Pi/2},10],CirclePoints[{1/2,Pi/2},6],CirclePoints[1/4,2]];
Graph[Range@Length@v,{},VertexCoordinates->v]

g=FindCorrespondingGraph[,v]//Quiet

RecognizeGraph@g

Configuration

GraphData["RubyGraph","Configuration"]

configurations=GraphData["RubyGraph","Graph","Configuration"]//StyleGraphs

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

GraphData["RubyGraph","Graph","CompleteKPartite"]//StyleGraphs

Bipartite

GraphData["RubyGraph","Bipartite"]

GraphData["RubyGraph","Graph","Bipartite"]//StyleGraphs

Bilateral

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

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

Circular

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

Circular construction All

GraphData["RubyGraph","Hamiltonian"]

GraphData["RubyGraph","HamiltonianCycles"]

GraphData["RubyGraph","HamiltonianCycleCount"]

GraphData["RubyGraph","LCF"]

GraphData["RubyGraph","LCFSignature"]

BilateralHamiltonianCycleQ[GraphData["RubyGraph"]]

FindBilateralHamiltonianCycle[GraphData["RubyGraph"]]

Planar

GraphData["RubyGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de

CrossingNumber[GraphData["RubyGraph"],Method->"WebCompute"]//AbsoluteTiming

DateString[]

Quit

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["RubyGraph","CN"]

(gs=GraphData["RubyGraph","Graph","MinimalCrossing"])//StyleGraphs

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["RubyGraph","RCN"]

GraphData["RubyGraph","Graph","MinimalRectilinearCrossing"]//StyleGraphs

MinimalCrossing construction Tutte

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

TutteGraphs=BilateralGraph[CenterGraph@IGLayoutTutte[g,"OuterFace"->#],"ReturnNonbilateralGraph"->True]&/@faces

TutteGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["RubyGraph","V"]])]
}&/@TutteGraphs;

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[TutteGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

LocalCrossingNumberWithEmbeddings=
{1+Max[Length/@DeleteCases[
Drop[AdjacencyList[#],GraphData["RubyGraph","V"]],
_Integer?(LessEqualThan[GraphData["RubyGraph","V"]]),
{2}
]],#}&/@TutteGraphs

TutteGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["RubyGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@TutteGraphs;

SymmetricEmbeddingsWithOrientations=DeleteCases[
TutteGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]

Select[SymmetricEmbeddingsWithOrientations[[All,1]],Count[OuterFace[#],_?(GreaterThan[GraphData["RubyGraph","V"]])]==0&]

MinimalCrossing construction Random 1k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["RubyGraph"],
1000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 5k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["RubyGraph"],
5000,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 20k [XXX s]

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

Length[diagrams=DistinctMinimalCrossingDiagramData[
GraphData["RubyGraph"],
20*^3,
"CrossingNumber"->cn,
EdgeShapeFunction->"BezierEdge"
]]//AbsoluteTiming

No crossing points on the boundary:

Length[diagrams0=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==0&],"Graph",{}]]//Timing

Take[diagrams0,UpTo[10]]

One crossing point on the boundary:

Length[diagrams1=Lookup[Select[diagrams,MinimalCrossingOuterCrossingCount[#]==1&],"Graph",{}]]//Timing

Take[diagrams1,UpTo[10]]

graphs=#["Graph"]&/@diagrams;

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[#]
}&/@graphs;//AbsoluteTiming

MaxMemoryUsed[]/10^9.GB

Symmetric embeddings:

Length[SymmetricMinimalEmbeddingData=DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]
]//AbsoluteTiming

SymmetricMinimalEmbeddingData

SymmetricMinimalEmbeddingGraphs=SymmetricMinimalEmbeddingData[[All,1]]

GraphCrossingCount/@SymmetricMinimalEmbeddingGraphs

GraphCrossingDiagram/@SymmetricMinimalEmbeddingGraphs

MinimalCrossing construction Random 1k [old]

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

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["RubyGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,1000}]//Union);//Timing

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["RubyGraph","V"]])]
}&/@RandomGraphs;

Small numbers of crossing points on the boundary:

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["RubyGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]/.g_Graph:>BilateralGraph[CenterGraph@g,"ReturnNonbilateralGraph"->True]

MinimalCrossing construction Random 20k [old]

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

(RandomGraphs=Table[MinimalCrossingGraph[GraphData["RubyGraph"],"CrossingNumber"->cn,RandomSeed->n],{n,20000}]//Union);//Timing

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["RubyGraph","V"]])]
}&/@RandomGraphs;

Small numbers of crossing points on the boundary:

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,0}:>g]

Cases[RandomGraphsWithOuterCrossingPointCounts,{g_,1}:>g]

Symmetric embeddings:

RandomGraphsWithSymmetryGroups={
#,
GraphSymmetryGroup[
#,
AdditionalData->Join[Table[True,GraphData["RubyGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

DeleteCases[
RandomGraphsWithSymmetryGroups,
{_,TransformationGroup[{}]}
]/.g_Graph:>BilateralGraph[CenterGraph@g,"ReturnNonbilateralGraph"->True]

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

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

MinimaRectilinearlLocalCrossing construction 1000

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

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

(gs=Select[Table[MinimalCrossingGraph[GraphData["RubyGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,1000}],!AdjacentCrossingsQ[#,cn]&]//Union)//Timing

MinimaRectilinearlLocalCrossing construction 20000

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

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

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

Circulant

GraphData["RubyGraph","Circulant"]

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

Circulant construction

GraphData["RubyGraph","NotationRules","Circulant"]

(ci=CirculantIndices[GraphData["RubyGraph"]])//AbsoluteTiming

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["RubyGraph","Line"]

LineGraphQ[GraphData["RubyGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["RubyGraph"]]

Halin

GraphData["RubyGraph","Halin"]

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

Rigid

GraphData["RubyGraph","Rigid"]

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

Flexible

GraphData["RubyGraph","Flexible"]

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

Voltage

gs=GraphData["RubyGraph","Graph","Voltage"]//StyleGraphs

VoltageGraphNotation/@gs//Column

Voltage construction

VoltageGraph[GraphData["RubyGraph"],"MaxResults"->20,Debug->True]

LCF

GraphData["RubyGraph","LCF"]

GraphData["RubyGraph","HamiltonianCycleCount"]

GraphData["RubyGraph","LCFSignature"]

GraphData["RubyGraph","LCFNotations"]

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

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

LCF compute FindBilateralHamiltonianCycle

BilateralHamiltonianCycleQ[GraphData["RubyGraph"],Debug->True]//AbsoluteTiming

(BilateralHamiltonianCycles=FindBilateralHamiltonianCycle[GraphData["RubyGraph"],Debug->True])//AbsoluteTiming

CircularEmbeddingFromHamiltonianCycle[GraphData["RubyGraph"],BilateralHamiltonianCycles[[1]]]

LCFNotation@%

LCF compute All

Divisors[GraphData["RubyGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

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

Length[gs=nicelcfgs]

gs

NiceLCFEmbeddingsWithSymmetryGroups={#,GraphSymmetryGroup[#]}&/@gs;

DeleteCases[NiceLCFEmbeddingsWithSymmetryGroups,{_,TransformationGroup[{Reflection[InfiniteLine[{0,0},{0,-1}]]}]}]

UnitDistance

GraphData["RubyGraph","UnitDistance"]

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

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["RubyGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["RubyGraph","VertexList"],GraphData["RubyGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Graph[VertexList[#],EdgeList[#],VertexCoordinates->GraphEmbedding[#]],MaxIterations->500]&/@Complement@@(GraphData["RubyGraph","Graph",#]&/@{"All","UnitDistance"})

TextGrid[
gdata=SortBy[
{#,DegenerateGraphEmbeddingQ[#],GraphSymmetryGroup[#],Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
],
Dividers->All
]

Cases[gdata,{__,0.}]

Symmetric:

gs=Cases[
gdata,
{g_,_,sym:Except[TransformationGroup[{}]],diff_}:>{
HighlightGraph[g1=BilateralGraph[CenterGraph[g],"ReturnNonbilateralGraph"->True],Flatten[FindCoincidentVertices[g1]]],
sym,
diff
}]

UnitDistance by refinement of stored LCF embeddings

FileByteCount[file=FileNameJoin[{$PropertiesDir,"LCFNotations-Nice",GraphNameToFileName["RubyGraph",".m.gz"]}]]

Tally[(lcf=Import[file])[[All,-1]]]//Timing//AbsoluteTiming

lcf=LCFNotation/@nicelcfgs;

gs=UnitDistanceEmbeddingRefine[LCFGraph@@#,MaxIterations->500]&/@lcf;//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

NondegenerateSymmetricGraphs=Select[
gdata,
And[
DegenerateGraphEmbeddingQ[#[[1]]]===False,
GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]
]&
]/.g_Graph:>CenterGraph[g]

UnitDistance by FindUnitDistanceEmbedding  (random embeddings returning on messages)

gs=FindUnitDistanceEmbedding[GraphData["RubyGraph"],50,"ReturnWithMessage"->True];//Quiet//Timing

gdata=SortBy[
{#,Max[Abs[EdgeLengths[#]-1]]}&/@gs,
Last
]

Cases[gdata,{g_?UnitDistanceGraphEmbeddingQ,_}:>g]

SymmetricEmebddings=Select[gdata,GraphSymmetryGroup[#[[1]]]=!=TransformationGroup[{}]&]

UnitDistance by FindUnitDistanceEmbedding  (random embedding waiting until UD is found)

FindUnitDistanceEmbedding[GraphData["RubyGraph"],"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance FindSubgraphUnitDistanceEmbedding

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

gs=UniqueEmbeddings[l]

gs={,,,,,,,,,,,,,,}

UnitDistance construction

UnitDistanceEmbeddingRefine[GraphicsToGraph@]

RecognizeGraph@%

Rotate[,Pi/2]

(soln=NSolve[{
x^2+y^2==1,
(x-1/2)^2+(y-h)^2==1,
c^2+(d-1)^2==1,
(c-a)^2+(b-d)^2==1,
(e-1/2)^2+(f-h)^2==1,
(a-i)^2+(b-j)^2==1,
(e-i)^2+(j-f)^2==1,
(i-k)^2+(l-j)^2==1,
(c+k)^2+(d-l)^2==1,
e^2+(f-z)^2==1,
k^2+(z+1-l)^2==1,
i==f},{h,z,x,y,a,b,c,d,e,f,i,j,k,l},Reals])//Timing

...ran overnight...

gs=Graph[Range[18],UndirectedEdge@@@{{1,2},{1,3},{1,4},{2,5},{2,8},{3,6},{3,9},{4,7},{4,10},{5,6},{5,11},{6,12},{7,11},{7,12},{8,13},{8,15},{9,14},{9,10},{10,15},{11,16},{12,14},{12,16},{13,17},{14,18},{15,18},{16,17},{17,18}},VertexCoordinates->({{0,z+1},{-k,l},{0,z},{k,l},{-i,j},{-e,f},{-c,d},{c,d},{e,f},{i,j},{-a,b},{-1/2,h},{0,1},{1/2,h},{a,b},{-x,y},{0,0},{x,y}}/.#),VertexLabels->"Name"]&/@soln

Matchstick

GraphData["RubyGraph","Matchstick"]

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

3D default

Graph3D[GraphData["RubyGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["RubyGraph","Dimension"]

(g=UnitDistanceEmbeddingRefine[
Graph3D[
GraphData["RubyGraph","VertexList"],
GraphData["RubyGraph","EdgeList"]
],
WorkingPrecision->25])//Timing

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["RubyGraph","Polyhedral"]

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

GraphData

GraphDataString

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

WriteString[file,GraphDataString[g2,
"StandardName"->"RubyGraph",
"Name"->"ruby graph",
TimeConstraint->10,
Debug->True,
"Skip"->{"Arboricity"}
]]

Close[file]