Analytic combinatorics of connected graphs

TL;DR

This paper applies analytic combinatorics to derive detailed asymptotic formulas for the enumeration of connected graphs and multigraphs with edges growing linearly with vertices, advancing beyond previous partial results.

Contribution

It provides a complete asymptotic expansion for the enumeration of connected graphs and multigraphs, using generating functions and analytic combinatorics techniques.

Findings

Derived formulas for asymptotic coefficients of connected graphs

Extended results to connected multigraphs

Achieved complete asymptotic expansions beyond first terms

Abstract

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, ie generating function manipulations, we derive a formula for the coefficients of the complete asymptotic expansion. The same result is derived for connected multigraphs.