Unicyclic Graph

Author

Eric W. Weisstein
May 16, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Counts

All

A236570

Total[Boole[Get[#]]]&/@FileNames["*.m",FileNameJoin[{$GraphDataDirectory,"Nodes","Unicyclic"}]]

Table[Total[Boole[UnicyclicGraphQ/@Graphs[n]]],{n,10}]//Timing

Connected

A001429

Table[Total[Boole[Get[FileNameJoin[{$GraphDataDirectory,"Nodes","Unicyclic","Unicyclic_"<>If[n<10,"0",""]<>ToString[n]<>".m"}]][[ConnectedGraphIndices[n]]]]],{n,10}]

Table[Total[Boole[UnicyclicGraphQ/@ConnectedGraphs[n]]],{n,10}]//Timing

Length/@Table[GraphData[{"Connected","Unicyclic"},n],{n,12}]

InverseEulerTransform[{0,0,1,2,5,13,33,89,240,657,1806,5026,13999,39260,110381,311465,880840,2497405,7093751,20187313,57537552,164235501,469406091,1343268050,3848223585,11035981711,31679671920,91021354454,261741776369}]

Graphics

All

GraphicsGrid[
Flatten[Partition[#,6,6,{1,1},{}]&/@Map[Show[GraphData[#,"Image"],PlotLabel->TextCell[Style[StringReplace[GraphData[#,"Name"],"\[Hyphen]"->"-"],Italic,FontFamily->"Times"],PageWidth->500/6,TextAlignment->Center]]&,Table[RecognizeGraph/@Select[Graphs[n],UnicyclicGraphQ],{n,3,5}],{2}],1],Dividers->{All,{Thick,Thick,Thick,True,Thick,True}},ImageSize->500,Alignment->Top
]

Connected

GraphicsGrid[
Flatten[Partition[#,6,6,{1,1},{}]&/@Map[Show[GraphData[#,"Image"],PlotLabel->TextCell[Style[StringReplace[GraphData[#,"Name"],"\[Hyphen]"->"-"],Italic,FontFamily->"Times"],PageWidth->500/6,TextAlignment->Center]]&,Table[RecognizeGraph/@Select[ConnectedGraphs[n],UnicyclicGraphQ],{n,3,6}],{2}],1],Dividers->{All,{Thick,Thick,Thick,Thick,True,True,Thick}},ImageSize->500,Alignment->Top
]

Enumeration

uni=GraphData["Unicyclic"]

smalluni=Cases[uni,_String]

GraphicsGrid[
Partition[#,UpTo[6]]&@Map[Show[GraphData[#,"Image"],PlotLabel->TextCell[Style[StringReplace[GraphData[#,"Name"],"\[Hyphen]"->"-"],Italic,FontFamily->"Times"],PageWidth->500/6,TextAlignment->Center]]&,smalluni],Dividers->All,ImageSize->500,Alignment->Top
]