Asymptotics for connected graphs and irreducible tournaments

TL;DR

This paper derives the complete asymptotic expansions for the probabilities that large labeled graphs are connected and large labeled tournaments are irreducible, providing combinatorial interpretations for the coefficients involved.

Contribution

It offers the first full asymptotic expansions for these probabilities along with combinatorial interpretations of the coefficients.

Findings

Explicit asymptotic formulas for large graphs and tournaments

Combinatorial interpretation of expansion coefficients

Enhanced understanding of connectivity and irreducibility probabilities

Abstract

We compute the whole asymptotic expansion of the probability that a large uniform labeled graph is connected, and of the probability that a large uniform labeled tournament is irreducible. In both cases, we provide a combinatorial interpretation of the involved coefficients.