Airy Phenomena and Analytic Combinatorics of Connected Graphs
Identifieur interne : 005A52 ( Hal/Checkpoint ); précédent : 005A51; suivant : 005A53Airy Phenomena and Analytic Combinatorics of Connected Graphs
Auteurs : Philippe Flajolet [France] ; Bruno Salvy [France] ; Gilles Schaeffer [France]Source :
Descripteurs français
Abstract
The enumeration of connected graphs has been so far dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis---specifically, the saddle-point method. The enumeration of connected graphs by excess derives from a simple saddle-point analysis; furthermore, a refined analysis based on coalescent saddle points yields an explicit connection with Airy functions.
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Hal:inria-00101063Le document en format XML
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<front><div type="abstract" xml:lang="en">The enumeration of connected graphs has been so far dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis---specifically, the saddle-point method. The enumeration of connected graphs by excess derives from a simple saddle-point analysis; furthermore, a refined analysis based on coalescent saddle points yields an explicit connection with Airy functions.</div>
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<idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LIX_EMC">Equipe modélisation combinatoire du laboratoire d'informatique de l'École Polytechnique</idno>
<idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno>
<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LABO-LORIA-SET" p="LORIA">LABO-LORIA-SET</idno>
<idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
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<notesStmt><note type="commentary">Rapport interne.</note>
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<sourceDesc><biblStruct><analytic><title xml:lang="en">Airy Phenomena and Analytic Combinatorics of Connected Graphs</title>
<author role="aut"><persName><forename type="first">Philippe</forename>
<surname>Flajolet</surname>
</persName>
<email>philippe.flajolet@inria.fr</email>
<idno type="halAuthorId">130534</idno>
<orgName ref="#struct-364717"></orgName>
<affiliation ref="#struct-2407"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Bruno</forename>
<surname>Salvy</surname>
</persName>
<email>Bruno.Salvy@inria.fr</email>
<idno type="idHal">bruno-salvy</idno>
<idno type="halAuthorId">132262</idno>
<orgName ref="#struct-364717"></orgName>
<affiliation ref="#struct-2407"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Gilles</forename>
<surname>Schaeffer</surname>
</persName>
<email>Gilles.Schaeffer@loria.fr</email>
<idno type="halAuthorId">129866</idno>
<orgName ref="#struct-441569"></orgName>
<affiliation ref="#struct-2345"></affiliation>
</author>
</analytic>
<monogr><idno type="reportNumber">A02-R-216 || flajolet02a</idno>
<imprint><biblScope unit="pp">25 p</biblScope>
<date type="datePub">2002</date>
</imprint>
</monogr>
</biblStruct>
</sourceDesc>
<profileDesc><langUsage><language ident="en">English</language>
</langUsage>
<textClass><keywords scheme="author"><term xml:lang="fr">asymptotics</term>
<term xml:lang="fr">connected graphs</term>
<term xml:lang="fr">énumération</term>
<term xml:lang="fr">asymptotique</term>
<term xml:lang="fr">graphes connexes</term>
</keywords>
<classCode scheme="halDomain" n="info.info-oh">Computer Science [cs]/Other [cs.OH]</classCode>
<classCode scheme="halTypology" n="REPORT">Reports</classCode>
</textClass>
<abstract xml:lang="en">The enumeration of connected graphs has been so far dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis---specifically, the saddle-point method. The enumeration of connected graphs by excess derives from a simple saddle-point analysis; furthermore, a refined analysis based on coalescent saddle points yields an explicit connection with Airy functions.</abstract>
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