Dominating Equivalent Graphs

Author

Eric W. Weisstein
December 3, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Counts

UniquePolynomialCount[n_,prop_]:=Module[
{
file=FileNameJoin[{$GraphDataDirectory,"Nodes",prop,prop<>"_"<>If[n<10,"0",""]<>ToString[n]<>".m"}],
lines
},
Count[Length/@Split[Sort[ReadList[file,"String",NullRecords->True]]],1]
]

A378516: Dominating unique

(unique=Table[UniquePolynomialCount[n,"DominationPolynomial"],{n,10}])//Timing

Length/@Table[GraphData["DominatingUnique",n],{n,8}]

Take[unique,8]-%

A378517: Dominating nonunique

nonunique=Table[SimpleGraphCount[n],{n,Length[unique]}]-unique

Length/@Table[GraphData["DominatingNonunique",n],{n,8}]

Take[nonunique,8]-%

Lists

<<MathWorld`Graphs`

CopolynomialGraphAssociation[n_,prop_,x_]:=Module[
{
file=FileNameJoin[{$GraphDataDirectory,"Nodes",prop,prop<>"_"<>If[n<10,"0",""]<>ToString[n]<>".m"}],
lines
},
SortBy[Select[GroupBy[MapIndexed[{ToExpression@#,#2[[1]]}&,ReadList[file,"String",NullRecords->True]],First->Last],Length[#]>1&],Length]
]

(rule4=CopolynomialGraphAssociation[4,"DominationPolynomial",x]//Normal)//Column

(rule5=CopolynomialGraphAssociation[5,"DominationPolynomial",x]//Normal)//Column

style=Directive[Black,FontFamily->"Times New Roman",14];

CopolynomialGraphicsRow[n_,rule_,x_]:=With[{gs=RecognizeGraph/@Graphs[n][[rule[[2]]]]},
Labeled[
GraphicsRow[StyleGraphs[
GraphData[#],
PlotLabel->If[MissingQ[not=GraphData[#,"Notation"]],
Style[GraphData[#,"Name"],style,Italic],
(* Else *)
Style[not/.char_String?(StringMatchQ[LetterCharacter]):>Style[char,Italic]/.Row[r_]:>Row[r," "],style]
]]&/@gs,Alignment->Top,ImageSize->300],
Style[rule[[1]]//TraditionalForm,style,16],
Top
]
]

Grid[
Partition[
Join[CopolynomialGraphicsRow[4,#,x]&/@rule4,CopolynomialGraphicsRow[5,#,x]&/@rule5],
UpTo[3]
],
Dividers->All,Alignment->Top]