Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae

TL;DR

This paper introduces a new systematic method using bivariate generating functions for deriving complete asymptotic expansions of dense graph families and 2-SAT formulae, with broad applicability and combinatorial insights.

Contribution

The authors develop a novel approach employing coefficient generating functions to obtain closed-form asymptotics for various dense graph structures and logical formulae, enhancing analytical flexibility.

Findings

Derived asymptotics for connected graphs and digraphs

Applied method to 2-SAT formulae and implication digraphs

Flexible framework accommodating structural variations

Abstract

We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of expansion coefficients into a special bivariate generating function, which we call a coefficient generating function. We show that coefficient generating functions possess certain general properties that make it possible to express asymptotics in a short closed form. Also, in most scenarios, we indicate a combinatorial meaning of the involved coefficients. Applications of our method include asymptotics of connected graphs, irreducible tournaments, strongly connected digraphs, 2-SAT formulae and contradictory strongly connected implication digraphs. Moreover, due to its flexibility, the method allows to treat a wide range of structural variations, including fixing the numbers of connected, irreducible, strongly connected and contradictory components, as well as source-like, sink-like and isolated ones, or adding weights and marking variables.