Metelsky Graphs

Author

Eric W. Weisstein
April 9, 2024

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

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

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

Figures

Metelsky

GraphicsGrid[Partition[Table[
Show[
StyleGraphs@GraphData[gn={"Metelsky",n}],
PlotLabel->TextCell[Style[
Column[
{
Row[{Subscript[Style["M",Italic],n]," == ",Subscript[Style["B",Italic],GraphData[gn,"NotationRules","Beineke"]]}],Style[GraphData[gn,"Name"],Italic]
},
Alignment->Center
],
Directive[14,Black,FontFamily->"Times"]
],
PageWidth->800/6,TextAlignment->Center]],{n,6}],UpTo[6]],ImageSize->800,Dividers->All,Alignment->Top]

Quit

MetelskyB

GraphicsGrid[Partition[Table[Show[StyleGraphs@GraphData[gn={"MetelskyB",n}],PlotLabel->TextCell[Column[{Style[Subscript[Style["B",Italic],n],Directive[14,Black,FontFamily->"Times"]],Style[GraphData[gn,"Name"],Directive[14,Black,Italic,FontFamily->"Times"]]},Alignment->Center],PageWidth->100,TextAlignment->Center]],{n,9}],UpTo[6]],ImageSize->700,Dividers->All,Alignment->Top]

MetelskyR

Quit

GraphicsGrid[Partition[Table[Show[StyleGraphs@GraphData[gn={"MetelskyR",n}],PlotLabel->TextCell[Column[{Style[Subscript[Style["R",Italic],n],Directive[14,Black,FontFamily->"Times"]],Style[GraphData[gn,"Name"],Directive[14,Black,Italic,FontFamily->"Times"]]},Alignment->Center],PageWidth->100,TextAlignment->Center]],{n,3}],UpTo[5]],ImageSize->700,Dividers->All,Alignment->Top]

Embeddings

<<MathWorld`Graphs`

GraphData["Metelsky*"]

GraphData["Metelsky"]

{"MetelskyB", 1} == {"Meteksly", 1} == {"Wheel", 6}

Names

GraphData[{"MetelskyB",1},"StandardName"]

GraphData[{"MetelskyB",1},"StandardNames"]

{"MetelskyB", 2} == {"Meteksly", 2} == "ClawGraph"

Names

GraphData[{"MetelskyB",2},"StandardName"]

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

{"MetelskyB", 3} == {"Meteksly", 3} == {"Dipyramid" ,3}

Names

GraphData[{"MetelskyB",3},"StandardName"]

GraphData[{"MetelskyB",3},"StandardNames"]

{"MetelskyB", 4} == {"Meteksly", 4} == {"King", {2, 3}}

Names

GraphData[{"MetelskyB",4},"StandardName"]

GraphData[{"MetelskyB",4},"StandardNames"]

{"MetelskyB", 5} == {"Meteksly", 5} == {"Beineke", 5}

Names

GraphData[{"MetelskyB",5},"StandardName"]

GraphData[{"MetelskyB",5},"StandardNames"]

{"MetelskyB", 6} == {"Meteksly", 6} == {"Wheel", 5}

Names

GraphData[{"MetelskyB",6},"StandardName"]

GraphData[{"MetelskyB",6},"StandardNames"]

{"MetelskyB", 7} == {"Meteksly", 7} == {"Beineke", 4}

Names

GraphData[{"MetelskyB",7},"StandardName"]

GraphData[{"MetelskyB",7},"StandardNames"]

{"MetelskyB", 8} == {"Meteksly", 8} == {"Beineke" ,7}

Names

GraphData[{"MetelskyB",8},"StandardName"]

GraphData[{"MetelskyB",8},"StandardNames"]

{"MetelskyB", 9} == {"Meteksly", 9} == {"Beineke", 8}

Names

GraphData[{"MetelskyB",9},"StandardName"]

GraphData[{"MetelskyB",9},"StandardNames"]

{"MetelskyR", 1} ==

Names

GraphData[{"MetelskyR",1},"StandardName"]

GraphData[{"MetelskyR",1},"StandardNames"]

{"MetelskyR", 2} == "DiamondGraph"

Names

GraphData[{"MetelskyR",2},"StandardName"]

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

{"MetelskyR", 3} == "KiteGraph"

Names

GraphData[{"MetelskyR",3},"StandardName"]

GraphData[{"MetelskyR",3},"StandardNames"]

Construction

MetelskyB

Metelsky, Y. and Tyshkevich, R.  "On Line Graphs of Linear 3-Uniform Hypergraphs."  J. Graph Theory 25, 243–251, 1997

MetelskyR

Metelsky, Y. and Tyshkevich, R.  "On Line Graphs of Linear 3-Uniform Hypergraphs."  J. Graph Theory 25, 243–251, 1997

v={{576,71},{527,68},{575,130},{529,175},{479,128},{372,130},{322,69},{273,128},{324,172},{114,170},{163,125},{66,126},{115,105},{116,68}};
MetelskyGraphsUnionR=Graph[Range@Length@v,UndirectedEdge@@@{e[12,14,11,10,12,11,13,12],{13,14},e[8,9,6,7,8,6],e[1,2,5,4,3,2],{5,3}},VertexCoordinates->v/100,VertexLabels->Automatic]

MetelskyGraphsR=ConnectedGraphComponents[MetelskyGraphsUnionR][[{2,3,1}]]

RecognizeGraph/@MetelskyGraphsR