Generalized Dodecagon

Author

Eric W. Weisstein
November 21, 2023

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

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

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

Table

Many of these are listed in van Dam and Haemers 2003

gd=Select[GraphData["GeneralizedPolygon"],First[GraphData[#,"NotationRules","GeneralizedPolygon"]]==12&]

TextGrid[Sort[
{
GraphData[#,"NotationRules","GeneralizedPolygon"][[2]],
GraphData[#,"V"],
GraphData[#,"Name"],
GraphData[#,"Notation"],
GraphData[#,"Spectrum"]//SpectrumForm
}&/@gd],Dividers->All,Alignment->{{Automatic,Decimal,{Left}}}]

GD(1,2)

GD(1,3) = Incidence graph of GH(3,3) = {Cage,{4,12}}

DistanceRegular.org

http://www.distanceregular.org/graphs/incidence-gh3 .3.html

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/incidence-gh33.am"]

RecognizeGraph@g

VertexCount[g]

"DistanceRegular"/.GraphData[{"Cage",{4,12}},"NotationRules"]

GraphData[{"Cage",{4,12}},"DistanceTransitive"]

GetGraphData[{"Cage",{4,12}},"File"]

GD(1,9)

Distance-transitive of 132860 nodes

Not on DistanceRegular.org

GD(2,1)

GraphData[{"GeneralizedDodecagon",{2,1}},"StandardNames"]

GraphData[{"GeneralizedDodecagon",{2,1}},"Names"]

GraphData[{"GeneralizedDodecagon",{2,1}},"Notation"]

GraphData[{"GeneralizedDodecagon",{2,1}},"NotationRules"]

GraphData[{"GeneralizedDodecagon",{2,1}},"Classes"]

GD(3,1)

Distance-transitive on 1456 nodes

{6,3,3,3,3,3;1,1,1,1,1,2}

Not on DistanceRegular.org

GraphData

GraphClassString

GraphClassString[gd]

GraphDataString

GD(2,1)

GraphData[{"Cage",{3,12}},"StandardNames"]

lg=LineGraph[Graph[{"Cage",{3,12}}]]

GraphPlot[lg]

RecognizeGraph[lg]

Print[GraphDataString[lg,
"Name"->"generalized dodecagon (2,1)",
"AlternateStandardNames"->{"GD2,1"},
"StandardName"->{"GeneralizedDodecagon",{2,1}},
"Classes"->{"DeterminedBySpectrum","GeneralizedPolygon","LineGraph"},
"NotationRules"->{"GeneralizedPolygon"->{12,{2,1}}},
"Information"->"GeneralizedDodecagon",
"Skip"->{"ChromaticPolynomial","ClawFree","Perfect"},
Debug->True,
TimeConstraint->60
]]