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| {{infobox graph
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| | name = Windmill graph
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| | image = [[Image:Windmill graph Wd(5,4).svg|220px]]
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| | image_caption = The Windmill graph Wd(5,4).
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| | vertices = ''(k-1)n+1''
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| | edges = ''nk(k−1)/2''
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| | automorphisms =
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| | girth = 3 if ''k > 2''
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| | diameter = 2
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| | radius = 1
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| | chromatic_number = ''k''
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| | chromatic_index = ''n(k-1)''
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| |notation = Wd(''k'',''n'')
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| | properties =
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| }}
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| In the [[mathematics|mathematical]] field of [[graph theory]], the '''windmill graph''' Wd(''k'',''n'') is an [[undirected graph]] constructed for ''k'' ≥ 2 and ''n'' ≥ 2 by joining ''n'' copies of the [[complete graph]] ''K<sub>k</sub>'' at a shared vertex. That is, it is a [[clique-sum|1-clique-sum]] of these complete graphs.<ref>Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic J. Combinatorics, DS6, 1-58, Jan. 3, 2007. [http://www.combinatorics.org/Surveys/ds6.pdf].</ref>
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| ==Properties== | | They simply call me [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=Dannie+Hayman&gs_l=news Dannie Hayman] however I don't truly like getting named like that. I am definitely fond of inside design but I haven't created a dime with it. [http://Www.guardian.co.uk/search?q=Pennsylvania Pennsylvania] has generally been my home. I get the job done as an information and facts officer. If you want to locate out more test out my web site: http://www.anpi.be/datatool/deeplinking/.svn/tmp/props/system/salomon-trailrunning.cfm<br><br>Feel free to visit my site; Salomon Trailrunning ([http://www.anpi.be/datatool/deeplinking/.svn/tmp/props/system/salomon-trailrunning.cfm similar web-site]) |
| It has ''(k-1)n+1'' vertices and ''nk(k−1)/2'' edges,<ref>{{MathWorld|urlname=WindmillGraph|title=Windmill Graph}}</ref> girth 3 (if ''k > 2''), radius 1 and diameter 2.
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| It has [[k-vertex-connected graph|vertex connectivity]] 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is ''(k-1)''-edge-connected. It is [[trivially perfect graph|trivially perfect]] and a [[block graph]].
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| ==Special cases==
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| By construction, the windmill graph Wd(3,''n'') is the [[friendship graph]] ''F<sub>n</sub>'', the windmill graph Wd(2,''n'') is the [[Star (graph theory)|star graph]] ''S<sub>n</sub>'' and the windmill graph Wd(3,2) is the [[butterfly graph]].
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| ==Labeling and colouring==
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| The windmill graph has [[chromatic number]] ''k'' and [[chromatic index]] ''n(k-1)''. Its [[chromatic polynomial]] can be deduced form the chromatic polynomial of the complete graph and is equal to
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| <math>\prod_{i=0}^{k-1}(x-i)^n.</math>
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| The windmill graph Wd(''k'',''n'') is proved not [[Graceful labeling|graceful]] if ''k'' > 5.<ref>K. M. Koh, D. G. Rogers, H. K. Teo, and K. Y. Yap, Graceful graphs: some further results and problems, Congr. Numer., 29 (1980) 559-571.</ref> In 1979, Bermond has conjectured that Wd(4,''n'') is graceful for all ''n'' ≥ 4.<ref>J.C. Bermond, Graceful graphs, radio antennae and French windmills, Graph Theory and Combinatorics, Pitman, London (1979) 18-37.</ref> This is known to be true for ''n'' ≤ 22.<ref>J. Huang and S. Skiena, Gracefully labeling prisms, Ars Combin., 38 (1994) 225-
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| 242.</ref> Bermond, Kotzig, and Turgeon proved that Wd(''k'',''n'') is not graceful when ''k'' = 4 and ''n'' = 2 or ''n'' = 3, and when ''k'' = 5 and ''m'' = 2.<ref>J. C. Bermond, A. Kotzig, and J. Turgeon, On a combinatorial problem of antennas in radioastronomy, in Combinatorics, A. Hajnal and V. T. Sos, eds., Colloq. Math. Soc. János Bolyai, 18, 2 vols. North-Holland, Amsterdam (1978) 135-149.</ref> The windmill Wd(3,''n'') is graceful if and only if ''n'' ≡ 0 (mod 4) or ''n'' ≡ 1 (mod 4).<ref>J.C. Bermond, A. E. Brouwer, and A. Germa, "Systèmes de triplets et différences associées", Problèmes Combinatoires et Théorie des Graphes, Colloq. Intern. du CNRS, 260, Editions du Centre Nationale de la Recherche Scientifique, Paris (1978) 35-38.</ref>
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| ==Gallery==
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| [[Image:Windmill graphs.svg|thumb|550px|center|Small windmill graphs.]]
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| {{-}}
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| == References ==
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| {{reflist}}
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| [[Category:Parametric families of graphs]]
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| [[Category:Perfect graphs]]
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