Rijeka Graph

Author

Eric W. Weisstein
April 30, 2025

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

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

<<MathWorld`Graphs`

Names

Renamed from "U42Graph216" to "RijekaGraph" in V14.3

GraphData["RijekaGraph","StandardName"]

GraphData["RijekaGraph","StandardNames"]

Primary

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

All

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

Inexact

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

Embedding types

GraphDataEmbeddings[GraphData["RijekaGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Degenerate

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

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

Perspective

GraphData["RijekaGraph","Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData["RijekaGraph","Polyhedron"]

Bilateral

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

Circular

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

Planar

GraphData["RijekaGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 1

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 2

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

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

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

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

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

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

MinimalCrossingNumber: "EmbeddingScheme" -> 3

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

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

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

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

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

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

(cn=CrossingNumber[GraphData["RijekaGraph"],"EmbeddingScheme"->3,TimeConstraint->2 86400])//AbsoluteTiming

MinimalCrossing construction

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

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

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

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

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

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

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

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["RijekaGraph","Circulant"]

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

Complement graph

GraphData["RijekaGraph","ComplementGraph","Name"]

Line graph

GraphData["RijekaGraph","Line"]

LineGraphQ[GraphData["RijekaGraph"]]//Timing

RootGraph

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

RootGraph[GraphData["RijekaGraph"]]

Halin

GraphData["RijekaGraph","Halin"]

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

Rigid

GraphData["RijekaGraph","Rigid"]

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

Flexible

GraphData["RijekaGraph","Flexible"]

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

LCF

GraphData["RijekaGraph","LCF"]

GraphData["RijekaGraph","HamiltonianCycleCount"]

GraphData["RijekaGraph","LCFSignature"]

GraphData["RijekaGraph","LCFNotations"]

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

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

LCF compute random

FindRandomLCFNotation[GraphData["RijekaGraph"],TimeConstraint->120]

FindRandomLCFNotation[GraphData["RijekaGraph"],TimeConstraint->3600]

FindRandomLCFNotation[GraphData["RijekaGraph"],TimeConstraint->86400]

LCF compute 10^4 [1 GB]

Tally[(lcf=LCF[GraphData["RijekaGraph"],10^4])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCF compute 10^5 [10 GB]

Tally[(lcf=LCF[GraphData["RijekaGraph"],10^5])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCF compute 10^6 [68 GB]

Tally[(lcf=LCF[GraphData["U42Graph216"],10^6])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

LCF compute 2*^6

Tally[(lcf=LCF[GraphData["U42Graph216"],2*^6])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

DeleteCases[lcf,{_,1}]

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

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

Length[gs]

LCF compute all

Divisors[GraphData["RijekaGraph","V"]]

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

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

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName["RijekaGraph",".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

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

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

Length[gs]

gs

UnitDistance

GraphData["RijekaGraph","UnitDistance"]

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

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

UnitDistance by refinement

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

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

UnitDistance by FindUnitDistanceEmbedding

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

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

Matchstick

GraphData["RijekaGraph","Matchstick"]

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

3D default

Graph3D[GraphData["RijekaGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["RijekaGraph","Polyhedral"]

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

Properties

CospectralGraphNames

GraphData["RijekaGraph","CospectralGraphNames"]

DeterminedBySpectrum

GraphData["RijekaGraph","DeterminedBySpectrum"]

DistanceRegular

GraphData["RijekaGraph","DistanceRegular"]

DistanceTransitive

GraphData["RijekaGraph","DistanceTransitive"]

IntersectionArray

GraphData["RijekaGraph","IntersectionArray"]

RegularParameters

GraphData["RijekaGraph","RegularParameters"]

Spectrum

SpectrumForm@GraphData["RijekaGraph","Spectrum"]//TraditionalForm

Construction

DistanceRegular.org

g=DistanceRegularGraph["rijeka.am"]

RecognizeGraph@g

RecognizeGraph@GraphComplement@g

StronglyRegularGraphQ[g]

RegularParameters[g]

IntersectionArray[g]

Through[{DistanceTransitiveGraphQ,DistanceRegularGraphQ}[g]]

CharacteristicPolynomialToSpectrum[CharacteristicPolynomial[AdjacencyMatrix[g],x],x]//SpectrumForm//TraditionalForm

Sage

https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html#sage.graphs.graph_generators.GraphGenerators.U42Graph216

from sage.features.gap import GapPackage
GapPackage("grape", spkg="gap_packages").is_present()
 G=graphs.U42Graph216()
G.edges()

data=;

g=CanonicalGraph[Graph[UndirectedEdge@@@Partition[StringCases[data,DigitCharacter..],2]]]

RegularParameters[g]//Timing

RecognizeGraph[g]

GraphData

GraphDataClassString

GraphDataClassString["RijekaGraph"]