Windmill Graph
Author
Eric W. Weisstein
Title
Windmill Graph
Description
The windmill graph D_n^((m)) is the graph obtained by taking m copies of the complete graph K_n with a vertex in common (Gallian 2011, p. 16). The case n=3 therefore corresponds to the Dutch windmill graph D_n^((m)). Precomputed properties of windmill graphs are implemented in the Wolfram Language as GraphData[{"Windmill", {m, n}}].
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z4am06/
DOI
https://notebookarchive.org/2019-07-0z4am06
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
112.91 kilobytes
Supplements
Rights
Redistribution rights reserved
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Windmill Graph
Windmill Graph
Author
Author
Eric W. Weisstein
July 7, 2018
July 7, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Figure
Figure
GraphicsGrid[Table[GraphPlot[WindmillGraph[m,n],PlotLabelSubsuperscript[W,n,m]],{m,2,5},{n,3,6}],ImageSize500]
Special cases
Special cases
TableForm[Table[{{m,n},RecognizeGraph[WindmillGraph[m,n]]},{m,2,5},{n,4,8}],TableDepth2]
{{2,4},{7,921}} | {{2,5},{}} | {{2,6},{}} | {{2,7},{}} | {{2,8},{}} |
{{3,4},{}} | {{3,5},{}} | {{3,6},{}} | {{3,7},{}} | {{3,8},{}} |
{{4,4},{}} | {{4,5},{}} | {{4,6},{}} | {{4,7},{}} | {{4,8},{}} |
{{5,4},{}} | {{5,5},{}} | {{5,6},{}} | {{5,7},{}} | {{5,8},{}} |
Construction
Construction
In[]:=
<<MathWorld`Graphs`
::shdw
::shdw
::shdw
In[]:=
Table[GraphData[{"Windmill",{n,3}}],{n,3,5}]
Out[]=
,
,
In[]:=
Table[GraphData[{"Windmill",{n,4}}],{n,3,5}]
Out[]=
,
,
In[]:=
GraphData[{"Windmill",{4,4}},"Graph","All"][[3]]
Out[]=
General
General
Code
Code
WindmillGraph[m_Integer,n_Integer,opts___?OptionQ]:=VertexDelete[System`Graph[Range[mn],EdgeList[GraphDisjointUnion@@Table[System`CompleteGraph[n],{m}]]/.Thread[1+Range[m-1]n1],opts],1+nRange[m-1]]
In[]:=
GraphData[{"Windmill",{2,4}}]
Out[]=
In[]:=
WindmillGraph[2,5]//RecognizeGraph
Out[]=
{Windmill,{2,5}}
Planar
Planar
In[]:=
WindmillGraphPlanar[n_,4]:=Module[{v=Flatten[Table[RotationMatrix[2πi/n].#&/@p,{i,0,n-1}],1],e={}},System`Graph[Range[Length[v]],UndirectedEdge@@@e,VertexCoordinatesv]]
In[]:=
WindmillGraphPlanar[3,4]
Out[]=
In[]:=
pp=Prepend[Flatten[Table[RotationMatrix[2πi/5].#&/@p,{i,0,4}],1],{0,0}]
In[]:=
e=Select[Subsets[Range[Length@pp],{2}],EuclideanDistance@@pp[[#]]
3.
&]In[]:=
g=System`Graph[Range[Length[pp]],UndirectedEdge@@@Join[e,{1,#}&/@{2,5,8,11,14},Table[{1,k},{k,3,15,3}],Table[{1,k+1},{k,3,15,3}]],VertexCoordinatespp,VertexLabels"Name"]
GraphData
GraphData
GraphDataString
GraphDataString
Do[Print[GraphDataString[WindmillGraph[m,n],"Name""("<>ToString[m]<>","<>ToString[n]<>")-windmill graph","StandardName"{"Windmill",{m,n}},"Classes"{"Windmill"},"NotationRules"{"Windmill"{m,n}},"Information""WindmillGraph","Skip"{"Perfect"},TimeConstraint120]],{m,2,5},{n,4,8}]
Cite this as: Eric W. Weisstein, "Windmill Graph" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z4am06
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