Taylor Graph

Author

Eric W. Weisstein
October 19, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

taylorfromrules=Select[GraphData["DistanceRegular"],MatchQ["DistanceRegular"/.GraphData[#,"NotationRules"],{{k_,μ_,1},{1,μ_,k_}}]&]

taylorclass=GraphData["Taylor"]

Complements[{taylorclass,taylorfromrules}]

TextGrid[Sort[{
GraphData[#,"V"],
GraphData[#,"Name"],
"DistanceRegular"/.GraphData[#,"NotationRules"]
}&/@Complement@@GraphData/@{"Taylor","Crown"}],Dividers->All,Alignment->{{Decimal,Left,Left}}]

ir[g_]:=GraphData[g,"Name"]<>"\n"<>StringReplace[StringTake[ToString["DistanceRegular"/.GraphData[g,"NotationRules"]],{2,-2}],"}, {":>"; "]

GraphicsGrid[Partition[Show[GraphData[#],PlotLabel->ir[#]]&/@SortBy[DeleteCases[l,{"Crown",_}],GraphData[#,"V"]&],3,3,{1,1},{}],ImageSize->500]

Construction

Taylor-5

lg=InduceSubgraph[g=Graph["IcosahedralGraph"],Rest[Neighborhood[g,1,1]]]

GraphData[RecognizeGraph[lg],"StandardNames"]

Taylor-9

GraphData[{"Johnson",{6,3}},"#"]

lg=InduceSubgraph[g=Graph[{"Tetrahedral",6}],Rest[Neighborhood[g,1,1]]]

FastIsomorphicQ[%25,Graph[{"Paley",9}]]

GraphData[RecognizeGraph[lg],"StandardNames"]

Taylor-13

InduceSubgraph[g=Graph[#],Rest[Neighborhood[g,1,1]]]&/@GraphData[28]

DistanceRegularQ[Graph[{"Barbell",14}]]

GraphData[{"Barbell",14},"Classes"]

lg=InduceSubgraph[g=Graph[XXX],Rest[Neighborhood[g,1,1]]]

GraphData[RecognizeGraph[lg],"StandardNames"]

The 2-graph is defined as follows (Taylor1977).  But how to get the graph from the 2-graph?

g=MakeGraph[Subsets[Range[8],{2}],Intersection[#1,#2]==={}&,Type->Undirected]

DistanceRegularQ[g]

IntersectionArray[g]

InduceSubgraph[g,Rest[Neighborhood[g,1,1]]]//RecognizeGraph

(c=GraphComplement[g])//DistanceRegularQ

IntersectionArray[c]

InduceSubgraph[c,Rest[Neighborhood[c,1,1]]]//RecognizeGraph

GraphData[{"Paley",15}]

g=MakeGraph[Subsets[Range[8],{2}],Length[Intersection[#1,#2]]==1&,Type->Undirected]

DistanceRegularQ[g]

IntersectionArray[GraphComplement[g]]

Locally Kneser K(6,2)

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/k62-taylor.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

GraphData[{"GeneralizedQuadrangle",{2,2}},"StandardNames"]

file=OpenWrite["~/locgq22.m"]

WriteString[file,GraphDataString[g,
"StandardName"->{"LocallyGeneralizedQuadrangle",{2,2}},
"Name"->"locally (2,2)\[Hyphen]generalized quadrangle graph",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

Close[file]

Locally 13-Paley

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/paley13-taylor.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

GetGraphData[{"NoncayleyTransitive",{28,32}},"File"]

Locally 17-Paley

g=DistanceRegularGraph["hhttp://www.distanceregular.org/graphdata/paley17-taylor.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

file=OpenWrite["~/locpayley17.m"]

WriteString[file,GraphDataString[g,
"StandardName"->{"LocallyPaley",17},
"Name"->"locally 17\[Hyphen]Paley graph",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

Close[file]

From HS 1

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/taylor-352a.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

file=OpenWrite["~/taylorhs.m"]

WriteString[file,GraphDataString[g,
"StandardName"->"TaylorGraphFromHigmanSims1",
"Name"->"Taylor graph from Higman\[Hyphen]Sims group 1",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

From HS 2

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/taylor-352b.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

WriteString[file,GraphDataString[g,
"StandardName"->"TaylorGraphFromHigmanSims2",
"Name"->"Taylor graph from Higman\[Hyphen]Sims group 2",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

Close[file]

From (Subscript[Co, 3]) 1

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/taylor-co3a.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

file=OpenWrite["~/taylorco3.m"]

WriteString[file,GraphDataString[g,
"StandardName"->"TaylorGraphFromConway1",
"Name"->"Taylor graph from Conway 3 group 1",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

From (Subscript[Co, 3]) 2

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/taylor-co3b.am"]

RecognizeGraph[g]

LocalGraph[g]//RecognizeGraph

WriteString[file,GraphDataString[g,
"StandardName"->"TaylorGraphFromConway2",
"Name"->"Taylor graph from Conway 3 group 2",
"Embeddings"->None,
"Information"->None,
"Classes"->{"DistanceRegular","DistanceTransitive","Noncirculant","Nongrid","Taylor"},
Debug->True,
TimeConstraint->10]]

Close[file]