Archimedean Dual Graph

Author

Eric W. Weisstein
October 25, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Nice planar embddings by Ed Pegg, Jr.

Properties

With[{props={"Name","V","M","|Aut|","EdgeTransitive","Hamiltonian","LCF","Planar","Regular","Traceable","VertexTransitive"}},
TextGrid[Prepend[GraphData[#,props]&/@GraphData["ArchimedeanDual"],props],Alignment->{{Left,Decimal,Decimal,Decimal,{Left}}},Dividers->Table[{Thick,Thick,{True},Thick},{2}]]
]

Graphics

GraphicsGrid[Partition[Show[GraphData[#]/.g_System`Graph:>SetProperty[g,{VertexSize->{"Scaled",.03},System`VertexStyle->Red,System`EdgeStyle->Black}],PlotLabel->TextCell[Style[GraphData[#,"Name"],FontFamily->"Times"],PageWidth->100,LineSpacing->{1,0},TextAlignment->Center]]&/@GraphData["ArchimedeanDual"],UpTo[5]],ImageSize->600,Spacings->{-20,Automatic},BaseStyle->{FontSlant->Italic},Dividers->All]

Line graphs

archdual=GraphData["ArchimedeanDual"]

archdualline=RecognizeGraph[System`LineGraph[GraphData[#]]]&/@archdual

archdualline=System`LineGraph[GraphData[#]]&/@archdual;

With[{props={"Name","LCF","Hamiltonian","Planar","Regular","Traceable"}},
TextGrid[Prepend[GraphData[#,props]&/@archlinedual,ToLowerCase[props]],Dividers->Table[{Thick,Thick,{True},Thick},{2}]]
]

archduallinenames=StringReplace[archdual,"Graph"->"LineGraph"]

Pick[archduallinenames,PlanarGraphQ/@archdualline]

Pick[archduallinenames,RegularGraphQ[#]&&HamiltonianGraphQ[#]&/@archdualline]