en>Mark viking: Added wl

2013-07-08T22:46:00Z

Added wl

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	~~The writer~~'~~s name~~ is ~~Andera~~ and ~~she believes it sounds quite great~~. ~~Invoicing~~ is ~~my occupation~~. ~~For many years she'~~s ~~been living in Kentucky but her spouse wants them~~ to ~~transfer~~. ~~To climb~~ is ~~something I really enjoy doing~~.<br><br>~~Feel free~~ to ~~visit my blog; online psychic reading~~, [~~http~~://~~formalarmour~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~26947/info~~/ http://~~formalarmour~~.~~com~~],		A '''random ''r''-regular graph''' is a [[graph (mathematics)\|graph]] selected from <math>\mathcal{G}_{n,r}</math>, which denotes the probability space of all [[regular graph\|''r''-regular graphs]] on ''n'' vertices, where 3 ≤ ''r'' < ''n'' and ''nr'' is even.<ref>[[Béla Bollobás]], ''Random Graphs'', 2nd edition, Cambridge University Press (2001), section 2.4: Random Regular Graphs</ref> It is therefore a particular kind of [[random graph]], but the regularity restriction significantly alters the properties that will hold, since most graphs are not regular.

			==Properties of random regular graphs==
			As with more general [[random graph]]s, it is possible to prove that certain properties of random ''r''-regular graphs hold [[almost surely]]. In particular, a random ''r''-regular graph is almost surely [[connectivity (graph theory)\|''r''-connected]].<ref>Bollobás, section 7.6: Random Regular Graphs</ref> In other words, although ''r''-regular graphs with connectivity less than ''r'' exist, the probability of selecting such a graph tends to 0 as ''n'' increases.

			If <math>\epsilon</math> > 0 is a positive constant, and ''d'' is the least integer satisfying

			<math>(r-1)^{d-1} \ge (2 + \epsilon)rn \ln n</math>

			then, almost surely, a random ''r''-regular graph has [[diameter (graph theory)\|diameter]] at most ''d''. There is also a (more complex) lower bound on the diameter of ''r''-regular graphs, so that almost all ''r''-regular graphs (of the same size) have almost the same diameter.<ref>Bollobás, section 10.3: The Diameter of Random Regular Graphs</ref>

			The distribution of the number of short cycles is also known: for fixed ''m'' ≥ 3, let ''Y''<sub>3</sub>,''Y''<sub>4</sub>,…,''Y''<sub>''m''</sub> be the number of cycles of lengths up to ''m''. Then the ''Y''<sub>''i''</sub> are asymptotically independent Poisson random variables with means<ref>Bollobás, section 2.4: Random Regular Graphs (Corollary 2.19)</ref>

			<math>\lambda_i=\frac{(r-1)^i}{2i}</math>

			==Algorithms for random regular graphs==
			It is non-trivial to implement the random selection of ''r''-regular graphs efficiently and in an unbiased way, since most graphs are not regular. The ''pairing model'' (also ''configuration model'') is a method which takes ''nr'' points, and partitions them into ''n'' buckets with ''r'' points in each of them. Taking a random matching of the ''nr'' points, and then contracting the ''r'' points in each bucket into a single vertex, yields an ''r''-regular graph or [[multigraph]]. If this object has no multiple edges or loops (i.e. it is a graph), then it is the required result. If not, a restart is required.<ref>N. Wormald, "Models of Random Regular Graphs," in ''Surveys in Combinatorics'', Cambridge University Press (1999), pp 239-298</ref>

			A refinement of this method was developed by [[Brendan McKay]] and Nicholas Wormald.<ref>B. McKay and N. Wormald, "Uniform Generation of Random Regular Graphs of Moderate Degree," ''Journal of Algorithms'', Vol. 11 (1990), pp 52-67: [http://cs.anu.edu.au/~bdm/papers/RandRegGen.pdf]</ref>

			==References==
			{{reflist}}

			[[Category:Random graphs]]
			[[Category:Regular graphs]]

en>Qetuth: more specific stub type

2011-12-21T15:23:21Z

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The writer's name is Andera and she believes it sounds quite great. Invoicing is my occupation. For many years she's been living in Kentucky but her spouse wants them to transfer. To climb is something I really enjoy doing.<br><br>Feel free to visit my blog; online psychic reading, [http://formalarmour.com/index.php?do=/profile-26947/info/ http://formalarmour.com],

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en>Mark viking: Added wl

en>Qetuth: more specific stub type