Doob Graph

Author

Eric W. Weisstein
February 10, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Images

l=GraphData["Doob"]

GraphicsGrid[{GraphPlot[Graph[#],Method->None,PlotLabel->"D("<>StringTake[ToString[Last[#]],{2,-2}]<>")",BaseStyle->{FontSlant->"Italic"}]&/@Take[GraphData["Doob"],3]},ImageSize->500,Spacings->-30]

LCF

Sort[GraphData[#,{"V","#"}]&/@l]

D(1,1)

Length[h=HamiltonianCycles[Graph[{"Doob",{1,1}}],All]]//Timing

Properties

GraphData["Doob","V"]

GraphData["Doob","Hamiltonian"]

GraphData["Doob","Integral"]

GraphData["Doob","Spectrum"]

FastIsomorphicQ[DoobGraph[1,3],DoobGraph[2,1]]

V/@{DoobGraph[1,3],DoobGraph[2,1]}

Enumeration

TableForm[Flatten[Table[{{m,n},RecognizeGraph[DoobGraph[m,n]]},{m,0,2},{n,2}],1],TableDepth->2]

Spectra

D (m,n)=H(n+2m,4)

D(1,1)==H(3,4)

IntegerSpectrum[DoobGraph[1,1]]//SpectrumForm//TraditionalForm

IntegerSpectrum[HammingGraph[3,4]]//SpectrumForm//TraditionalForm

RegularParameters[Graph[#]]&/@{{"Doob",{1,1}},{"Hamming",{3,4}}}

FastIsomorphicQ@@Graph/@{{"Doob",{1,1}},{"Hamming",{3,4}}}

D(1,2)==H(4,4)

IntegerSpectrum[DoobGraph[1,2]]//SpectrumForm//TraditionalForm

IntegerSpectrum[HammingGraph[4,4]]//SpectrumForm//TraditionalForm

RegularParameters[Graph[#]]&/@{{"Doob",{1,2}},{"Hamming",{4,4}}}

FastIsomorphicQ@@Graph/@{{"Doob",{1,2}},{"Hamming",{4,4}}}

D(2,1)==H(5,4)

IntegerSpectrum[DoobGraph[2,1]]//SpectrumForm//TraditionalForm

IntegerSpectrum[HammingGraph[5,4]]//SpectrumForm//TraditionalForm

GraphData

GraphClassString

GraphClassString[GraphData["Doob"]]

GraphDataString

(n+2m,4)

new=Select[Flatten[Table[{d,q},{d,10},{q,5}],1],4^({2,1}.#)<2100&]

file=OpenWrite["doob.m"]

Function[{d,q},WriteString[file,GraphDataString[
DoobGraph[d,q],
"StandardName"->{"Doob",{d,q}},
"Name"->"("<>ToString[d]<>","<>ToString[q]<>")-Doob graph",
"Information"->"DoobGraph",
"Classes"->{"DistanceRegular","Doob"},
"NotationRules"->{"Doob"->{d,q}},
Debug->All,
TimeConstraint->60
]]]@@@new

Close[file]