Lalonde Graph

Author

Eric W. Weisstein
June 2, 2026

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

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

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

Source

https://arxiv.org/pdf/2311.08194  
Smallest known example separating rank-1 and rank-2 quantum chromatic behavior. 

Highly asymmetric. I don’t care for it...

LalondeGraph=ImportString["TX_ac~QhaBO_TDaO@dDewW_gCd?WWI_c[?lg","Graph6"]

Through[{VertexCount,EdgeCount}[LalondeGraph]]

Figure

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

Embeddings

<<MathWorld`Graphs`

Names

GraphData["LalondeGraph","StandardName"]

GraphData["LalondeGraph","StandardNames"]

HouseOfGraphs

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

HouseOfGraphsLookup[hogid]

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

Wikidata

GraphData["LalondeGraph","WikidataID"]

GraphData["LalondeGraph","WikidataID","URL"]

Primary

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

All

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

CurvedEdge

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

Inexact

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

Degenerate

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

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

Embeddings

GraphDataEmbeddings[GraphData["LalondeGraph"]]

Default embedding

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

Configuration

GraphData["LalondeGraph","Configuration"]

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

GeneralizedPetersen

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

IGraph

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

CompleteKPartite

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

Bipartite

GraphData["LalondeGraph","Bipartite"]

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

Bilateral

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

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

Circular

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

Circular construction 1*^4 [0.03 + 8.8 s]

Length[hc=FindHamiltonianCycle[GraphData["LalondeGraph"],1*^4][[All,All,1]]]//Timing

Length[hcsym=SymmetricHamiltonianCycles[GraphData["LalondeGraph"],hc,ProgressReporting->True]]//Timing

GroupBy[SymmetricHamiltonianCycleGraph[GraphData["LalondeGraph"],#]&/@hcsym,GraphSymmetryGroup]

Circular construction 1*^5 [0.3 +  s]

Length[hc=FindHamiltonianCycle[GraphData["LalondeGraph"],1*^5][[All,All,1]]]//Timing

Length[hcsym=SymmetricHamiltonianCycles[GraphData["LalondeGraph"],hc,ProgressReporting->True]]//Timing

GroupBy[SymmetricHamiltonianCycleGraph[GraphData["LalondeGraph"],#]&/@hcsym,GraphSymmetryGroup]

gs=BilateralGraph@SymmetricHamiltonianCycleGraph[GraphData["LalondeGraph"],#]&/@Take[hcsym,10]

Circular construction All

GraphData["LalondeGraph","Hamiltonian"]

GraphData["LalondeGraph","HamiltonianCycles"]

GraphData["LalondeGraph","HamiltonianCycleCount"]

GraphData["LalondeGraph","LCF"]

GraphData["LalondeGraph","LCFSignature"]

Length[hc=FindHamiltonianCycle[GraphData["LalondeGraph"],All][[All,All,1]]]//Timing

Length[hcsym=SymmetricHamiltonianCycles[GraphData["LalondeGraph"],hc,ProgressReporting->True]]//Timing

GroupBy[SymmetricHamiltonianCycleGraph[GraphData["LalondeGraph"],#]&/@hcsym,GraphSymmetryGroup]

Planar

GraphData["LalondeGraph","Planar"]

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

MinimalCrossingNumber crossings.uos.de [failed]

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

DateString[]

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

MinimalCrossing

GraphData["LalondeGraph","CN"]

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

GraphCrossingCount/@gs

GraphCrossingDiagram/@gs

MinimalRectilinearCrossing

GraphData["LalondeGraph","RCN"]

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

MinimalCrossing construction Tutte

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

(g=MinimalCrossingGraph[GraphData["LalondeGraph"],"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["LalondeGraph","V"]])]
}&/@TutteGraphs;

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

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

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

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

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

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

MinimalCrossing construction Random 1k [XXX s]

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

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

No crossing points on the boundary:

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

One crossing point on the boundary:

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

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["LalondeGraph"],TimeConstraint->1])//AbsoluteTiming

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

No crossing points on the boundary:

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

One crossing point on the boundary:

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

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["LalondeGraph"],TimeConstraint->1])//AbsoluteTiming

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

No crossing points on the boundary:

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

One crossing point on the boundary:

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

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["LalondeGraph"],TimeConstraint->1])//AbsoluteTiming

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["LalondeGraph","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["LalondeGraph","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["LalondeGraph"],TimeConstraint->1])//AbsoluteTiming

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

RandomGraphsWithOuterCrossingPointCounts={
#,
Count[OuterFace[#],_?(GreaterThan[GraphData["LalondeGraph","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["LalondeGraph","V"]],Table[False,cn]],
SameTest->Equal
]
}&/@RandomGraphs;

MaxMemoryUsed[]/10^9.GB

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

MinimalLocalCrossing

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

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

MinimalRectilinearLocalCrossing

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

GraphData["LalondeGraph","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["LalondeGraph"],TimeConstraint->10])//AbsoluteTiming

(gs=Select[Table[MinimalCrossingGraph[GraphData["LalondeGraph"],"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["LalondeGraph"],TimeConstraint->10])//AbsoluteTiming

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

Circulant

GraphData["LalondeGraph","Circulant"]

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

Circulant construction

GraphData["LalondeGraph","NotationRules","Circulant"]

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

CirculantGraph1@@@ci

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

CirculantGraph1@@@%

Complement graph

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

Line graph

GraphData["LalondeGraph","Line"]

LineGraphQ[GraphData["LalondeGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["LalondeGraph"]]

Halin

GraphData["LalondeGraph","Halin"]

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

Rigid

GraphData["LalondeGraph","Rigid"]

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

Flexible

GraphData["LalondeGraph","Flexible"]

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

Voltage

GraphData["LalondeGraph","Graph","Voltage"]//StyleGraphs

Voltage construction

gs=VoltageGraph[GraphData["LalondeGraph"],"MaxResults"->20,Debug->True]

Length[gs]

RecognizeGraph/@gs

VoltageGraphNotation/@gs//Column

EmbeddingClasses/@gs//Column

GraphSymmetryGroup/@gs//Column

LCF compute ssham_lcf

l1={
{"LalondeGraph",{0,sig}}
};

PropertyReplace[l1,"LCFSignature"]

RenameGraphFiles[]

ReverseSort@Tally[(lcf=Import[FileNameJoin[{$GraphDataDirectory,"Properties","LCFNotations-Nice",GraphNameToFileName["LalondeGraph",".m.gz"]}]])[[All,2]]]

gs=NiceLCFGraphs[lcf,3,2]

Length[gs]

nicelcf=LCFNotation/@gs

l2={
{"LalondeGraph",{0,Missing["External",nicelcf]}}
}

PropertyReplace[l2,"LCFNotations"]

RenameGraphFiles[]

Length[BilateralOrder1LCF=Get[FileNameJoin[{"/","Volumes","eww","mma","Graphs","Properties","LCFNotations-Nice",GraphNameToFileName["LalondeGraph","_bilateral-order-1-reserve.log"]}]]]

BilateralLCFGraph@@@BilateralOrder1LCF//Length//Timing

BilateralLCFGraph@@@Take[BilateralOrder1LCF,10]

Length[bilat=SelectBilateralLCFs[BilateralOrder1LCF,"ProgressEvery"->500,Debug->True]]//Timing

LCF

GraphData["LalondeGraph","LCF"]

GraphData["LalondeGraph","HamiltonianCycleCount"]

GraphData["LalondeGraph","LCFSignature"]

GraphData["LalondeGraph","LCFNotations"]

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

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

UnitDistance

GraphData["LalondeGraph","UnitDistance"]

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

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

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

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

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Graph[VertexList[#],EdgeList[#],VertexCoordinates->GraphEmbedding[#]],MaxIterations->500]&/@Complement@@(GraphData["LalondeGraph","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 FindUnitDistanceEmbedding  (random embeddings returning on messages)

gs=FindUnitDistanceEmbedding[GraphData["LalondeGraph"],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["LalondeGraph"],"ReturnWithMessage"->False]//Quiet//Timing

Matchstick

GraphData["LalondeGraph","Matchstick"]

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

3D default

Graph3D[GraphData["LalondeGraph","Edges"]]

3D construction

3D All

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

3D UnitDistance construction Random

GraphData["LalondeGraph","Dimension"]

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

EdgeLengths[g]

UnitDistanceGraphEmbeddingQ[g]

3D UnitDistance

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

3D Polyhedron

GraphData["LalondeGraph","Polyhedral"]

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

GraphData

GraphDataString

<<MathWorld`Graphs`

outstream=OpenWrite["~/LalondeGraph.m"]

LalondeGraph=ImportString["TX_ac~QhaBO_TDaO@dDewW_gCd?WWI_c[?lg","Graph6"]

WriteString[
outstream,
GraphDataString[
LalondeGraph,
"StandardName"->"LalondeGraph",
"Name"->"Lalonde graph"
]
];

Close[outstream]