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Ballots, queues and random graphs

Published online by Cambridge University Press:  14 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Case Western Reserve University
*
∗Postal address: Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA.
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Abstract

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

Keywords

CONNECTEDNESSEXACT AND LIMIT DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 103 - 112
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Research supported by National Science Foundation Grant DMS-85-10073 and Office of Naval Research Grant N0001485-K-009.

References

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