Tadpole Graph

Author

Eric W. Weisstein
March 3, 2018

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

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

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

Embeddings

Old

GraphicsGrid[Table[GraphPlot[TadpoleGraph[m,n],Method->None,PlotLabel->Subscript[T,Sequence@@{m,n}]],{m,3,6},{n,4}],ImageSize->500,Spacings->{0,-50}]

Graphs.m

Default

GraphicsGrid[Table[TadpoleGraph[m,n,PlotLabel->Subscript[T,Sequence@@{m,n}]],{m,3,6},{n,4}],ImageSize->500,Spacings->{10,-50},Dividers->All]//StyleGraphs

"UnitDistance"

GraphicsGrid[Table[TadpoleGraph[m,n,GraphLayout->"UnitDistance",PlotLabel->Subscript[T,Sequence@@{m,n}]],{m,3,6},{n,4}],ImageSize->500,Spacings->{10,-50},Dividers->All]//StyleGraphs

GraphData

Primary

GraphicsGrid[Table[Show[GraphData[{"Tadpole",{m,n}}]//StyleGraphs,GraphLayout->"UnitDistance",PlotLabel->Subscript[T,Sequence@@{m,n}]],{m,3,7},{n,5}],ImageSize->500,Spacings->{10,10},Dividers->All]

"UnitDistance"

GraphicsGrid[Table[Show[GraphData[{"Tadpole",{m,n}},"Graph","UnitDistance"][[1]]//StyleGraphs,GraphLayout->"UnitDistance",PlotLabel->Subscript[T,Sequence@@{m,n}]],{m,3,7},{n,5}],ImageSize->500,Spacings->{10,10},Dividers->All]

Special cases

TextGrid[DeleteCases[{"Tadpole"/.GraphData[#,"NotationRules"],#}&/@GraphData["Tadpole"],{_,{"Pan"|"Tadpole",_}}],Dividers->All]

Counts

Edges

Subtract@@@({{n,k}=("Tadpole"/.GraphData[#,"NotationRules"]);k+n,GraphData[#,"M"]}&/@GraphData["Tadpole"])//Union

{n,k}=.;

(k-1)n+n-1//Simplify

Vertices

Subtract@@@({{n,k}=("Tadpole"/.GraphData[#,"NotationRules"]);k+n,GraphData[#,"V"]}&/@GraphData["Tadpole"])//Union

GraphData

GraphClassString

GraphClassString[GraphData["Tadpole"]]

GraphDataString

Do[Print[GraphDataString[TadpoleGraph[n,k],
"StandardName"->{"Tadpole",{n,k}},
"AlternateStandardNames"->Switch[{n,k},{3,1},{"Lollipop",{3,1}},{3,2},{"Lollipop",{3,2}},{3,3},{"Lollipop",{3,3}},{3,4},{"Lollipop",{3,4}},{3,5},{"Lollipop",{3,5}},{4,1},{5,16},{4,2},{6,54},{4,3},{7,191},{4,4},{8,2265},{5,1},{6,63},{5,2},{7,374},{5,3},{8,2263},{6,1},{7,212},{6,2},{8,1290},{7,1},{8,2319},_,{}],
"Name"->"("<>ToString[n]<>","<>ToString[k]<>")-tadpole graph",
"Skip"->{"Perfect"},
"Classes"->If[n==3,{"Lollipop","Tadpole"},{"Tadpole"}],
TimeConstraint->120
]
],{n,3,7},{k,5}]

Add UnitDistance

l=Flatten[Table[{gn={"Tadpole",{m,n}},{0,Append[GraphData[gn,"Embeddings"],ToCommonEdges[TadpoleGraph[m,n,GraphLayout->"UnitDistance"],GraphData[gn]]]//Simplify}},{m,{3,4,5,7}},{n,2,5}],1]

PropertyReplace[l,"Embeddings","11.3"]

RenameGraphFiles[]

l=Table[{gn={"Pan",n},{0,Append[GraphData[gn,"EmbeddingClasses"],{"Integral","MinimalCrossing","MinimalIntegral","MinimalPlanarIntegral","MinimalRectilinearCrossing","Planar","UnitDistance"}]}},{n,5,20}]

l=Flatten[Table[{gn={"Tadpole",{m,n}},{0,Append[GraphData[gn,"EmbeddingClasses"],{"Integral","MinimalCrossing","MinimalIntegral","MinimalPlanarIntegral","MinimalRectilinearCrossing","Planar","UnitDistance"}]}},{m,{3,4,5,7}},{n,2,5}],1]

PropertyReplace[l,"EmbeddingClasses","11.3"]

RenameGraphFiles[]