Abstract

One of the classical topics in discrete mathematics and in computer science is the computation of the transitive closure and reduction of a directed graph. We provide a complete analysis of the random variables counting the size of the transitive closure and reduction in random acyclic digraphs. In particular we present closed forms for their probability distribution and moments, showing that they involve functions such as the q--binomial coefficients and the q--Stirling numbers of the second kind. In general it turns out that a pure numbertheoretical notation, the q--notation, represents the perfect tool to deal with these kinds of problems. In this context we speak therefore about q--distributions. 1 Introduction Graphs are probably the most common "abstract" structure in computer science and because of their flexibility they provide an adequate model for a variety of problems. Any system consisting of a set V of discrete states (called nodes or vertices) and a set E of relations b...

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