Asymptotics for the Probability of Connectedness and the Distribution of Number of Components
-
Published:2000-05-30
Issue:1
Volume:7
Page:
-
ISSN:1077-8926
-
Container-title:The Electronic Journal of Combinatorics
-
language:
-
Short-container-title:Electron. J. Combin.
Author:
Bell Jason P.,Bender Edward A.,Cameron Peter J.,Richmond L. Bruce
Abstract
Let $\rho _n$ be the fraction of structures of "size" $n$ which are "connected"; e.g., (a) the fraction of labeled or unlabeled $n$-vertex graphs having one component, (b) the fraction of partitions of $n$ or of an $n$-set having a single part or block, or (c) the fraction of $n$-vertex forests that contain only one tree. Various authors have considered $\lim \rho _n$, provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair $(\liminf \rho _{n},\;\limsup \rho _{n})$ in these cases. Only in the convergent case can one have $0 < \lim \rho _{n} < 1$. We study the existence of $\lim \rho _{n}$ in this case.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
8 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Unlabelled Gibbs partitions;Combinatorics, Probability and Computing;2019-11-04
2. Gibbs partitions: The convergent case;Random Structures & Algorithms;2018-02-27
3. Asymptotic Properties of Some Minor-Closed Classes of Graphs;Combinatorics, Probability and Computing;2014-07-09
4. Random Graphs from a Minor-Closed Class;Combinatorics, Probability and Computing;2009-07
5. Random cubic planar graphs;Random Structures and Algorithms;2006