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← Older revision		Revision as of 22:33, 31 October 2013
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	~~Yoshiko is her name but she doesn~~'~~t like when people use her~~ complete ~~name~~. ~~Some time~~ in the ~~past I chose to live in Arizona but I require to move for my family members~~. ~~His day job is~~ a ~~monetary officer but he plans on altering it~~. ~~Playing crochet is some thing that I~~'~~ve done for many years~~.<br><br>~~Have~~ a ~~look at my blog post~~ [http://~~web321~~.~~sv09~~.~~net-housting~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~8837 extended auto warranty~~]		[[Image:Cayley's formula 2-4.svg\|thumb\|The complete list of
			all trees on 2,3,4 labeled vertices: <math>2^{2-2}=1</math> tree with 2 vertices,
			<math>3^{3-2}=3</math> trees with 3 vertices and <math>4^{4-2}=16</math>
			trees with 4 vertices.]]

			In [[mathematics]], '''Cayley's formula''' is a result in [[graph theory]] named after [[Arthur Cayley]]. It states that for any positive integer ''n'', the number of [[Tree (graph theory)\|trees]] on ''n'' labeled [[vertex (graph theory)\|vertices]] is <math>n^{n-2}</math>.

			The formula equivalently counts the number of [[spanning tree]]s of a [[complete graph]] with labeled vertices
			{{OEIS\|id=A000272}}.

			==Proof==
			Many remarkable proofs of Cayley's tree formula are known.<ref>{{Cite book
			\| last1 = Aigner \| first1 = Martin \| author1-link = Martin Aigner
			\| last2 = Ziegler \| first2 = Günter M. \| author2-link = Günter M. Ziegler
			\| pages = 141–146
			\| publisher = [[Springer-Verlag]]
			\| title = [[Proofs from THE BOOK]]
			\| year = 1998}}</ref>
			One classical proof of the formula uses [[Kirchhoff's matrix tree theorem]]. [[Prüfer sequence]]s yield a [[bijective proof]] of Cayley's formula. Another bijective proof, by [[André Joyal]], finds a one-to-one transformation between ''n''-node trees with two distinguished nodes and maximal directed [[pseudoforest]]s.
			A proof by [[double counting (proof technique)\|double counting]] due to Jim Pitman [[Double counting (proof technique)#Counting trees\|applies to trees]].

			== History ==
			The formula was first discovered by [[Carl Wilhelm Borchardt]] in 1860, and proved via a [[determinant]].<ref>{{cite journal \|last=Borchardt \| first = C. W. \| authorlink = Carl Wilhelm Borchardt \|year=1860 \|title=Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung \|journal=Math. Abh. der Akademie der Wissenschaften zu Berlin \|pages=1–20}}</ref> In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices.<ref>{{cite journal \|first=A.\|last=Cayley\|authorlink=Arthur Cayley \|url=http://books.google.com/?id=M7c4AAAAIAAJ&pg=PA26 \|title=A theorem on trees \|journal=Quart. J. Math \|volume=23 \|year=1889 \|pages=376–378}}</ref> Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.

			==See also==
			*[[Kirchhoff's theorem]], a formula for the number of spanning trees in an arbitrary graph involving the [[determinant]] of a [[matrix (mathematics)\|matrix]].

			==References==
			{{reflist}}

			[[Category:Articles containing proofs]]
			[[Category:Trees (graph theory)]]

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Equilibrium

2012-08-29T01:14:49Z

Repairing links to disambiguation pages - You can help! - Equilibrium

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Yoshiko is her name but she doesn't like when people use her complete name. Some time in the past I chose to live in Arizona but I require to move for my family members. His day job is a monetary officer but he plans on altering it. Playing crochet is some thing that I've done for many years.<br><br>Have a look at my blog post [http://web321.sv09.net-housting.de/index.php?mod=users&action=view&id=8837 extended auto warranty]

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