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| [[File:Cayley's formula 2-4.svg|thumb|The complete list of | | Nestor is the name my [http://www.Dmv.org/buy-sell/auto-warranty/extended-warranty.php parents] gave me but I don't like extended auto warranty when people use my extended [http://muehle-kruskop.de/index.php?mod=users&action=view&id=22344 auto warranty] full title. Interviewing is what she does. Arizona has usually been my residing place but my wife wants us to transfer. One of my preferred hobbies is camping and now I'm trying to earn cash with extended [http://www.cars.com/go/advice/Story.jsp?section=buy&story=usedWarranty&subject=warranty auto warranty] it.<br><br>Also visit my blog post: extended [http://Www.Nbcnews.com/id/23147777/ns/business-consumer_news/t/auto-warranty-firms-launch-sleazy-scam/ extended auto] warranty [http://www.obsession-circle.de/index.php?mod=users&action=view&id=24823 car warranty] ([http://www.itcomplianceexperts.com/UserProfile/tabid/165/userId/36272/Default.aspx just click the following document]) |
| all free trees on 2,3,4 labeled vertices: <math>2^{2-2}=1</math> tree with 2 vertices,
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| <math>3^{3-2}=3</math> trees with 3 vertices and <math>4^{4-2}=16</math>
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| trees with 4 vertices.]]
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| In [[combinatorics]], an area of [[mathematics]], '''graph enumeration''' describes a class of [[combinatorial enumeration]] problems in which one must count [[undirected graph|undirected]] or [[directed graph]]s of certain types, typically as a function of the number of vertices of the graph.<ref>{{cite book
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| |last1 = Harary | first1 = Frank | author1-link = Frank Harary | first2 = Edgar M. | last2 = Palmer | year = 1973| title = Graphical Enumeration | publisher = [[Academic Press ]] | id = ISBN 0-12-324245-2}}</ref> The pioneers in this area of mathematics were [[George Pólya|Pólya]], <ref> Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68 (1937), 145-254 </ref> [[Arthur Cayley |Cayley]] <ref>{{acad|id=CLY838A|name=Cayley, Arthur}}</ref> and [[John Howard Redfield | Redfield]].<ref>The theory of group-reduced distributions. American J. Math. 49 (1927), 433-455.</ref>
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| In some graphical enumeration problems, the vertices of the graph are considered to be ''labeled'' in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph. In general, labeled problems tend to be easier to solve than unlabeled problems.<ref>Harary and Palmer, p. 1.</ref> As with combinatorial enumeration more generally, the [[Pólya enumeration theorem]] is an important tool for dealing with symmetries such as this.
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| Some important results in this area include the following.
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| *The number of labeled ''n''-vertex undirected graphs is 2<sup>''n''(''n'' − 1)/2</sup>.<ref>Harary and Palmer, p. 3.</ref>
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| *The number of labeled ''n''-vertex directed graphs is 2<sup>''n''(''n'' − 1)</sup>.<ref>Harary and Palmer, p. 5.</ref>
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| *The number ''C<sub>n</sub>'' of [[connected graph|connected]] labeled ''n''-vertex undirected graphs satisfies the [[recurrence relation]]<ref>Harary and Palmer, p. 7.</ref>
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| ::<math>C_n=2^{n\choose 2} - \frac{1}{n}\sum_{k=1}^{n-1} k{n\choose k} 2^{n-k\choose 2} C_k.</math>
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| :from which one may easily calculate, for ''n'' = 1, 2, 3, ..., that the values for ''C<sub>n</sub>'' are | |
| ::1, 1, 4, 38, 728, 26704, 1866256, ...{{OEIS|id=A001187}}
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| *The number of labeled ''n''-vertex [[free tree]]s is ''n''<sup>''n'' − 2</sub> ([[Cayley's formula]]).
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| *The number of unlabeled ''n''-vertex [[caterpillar tree|caterpillars]] is<ref>{{citation
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| | last1 = Harary | first1 = Frank | author1-link = Frank Harary
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| | last2 = Schwenk | first2 = Allen J.
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| | issue = 4
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| | journal = Discrete Mathematics
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| | pages = 359–365
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| | title = The number of caterpillars
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| | volume = 6
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| | year = 1973}}.</ref>
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| ::<math>2^{n-4}+2^{\lfloor (n-4)/2\rfloor}.</math>
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| ==References==
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| {{reflist|2}}
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| [[Category:Graph enumeration]]
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| [[Category:Enumerative combinatorics]]
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| {{combin-stub}}
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