Counting connected graphs with large excess

TL;DR

This paper derives the complete asymptotic expansion for counting connected graphs with a linear number of edges relative to vertices, advancing beyond the first-term approximation using analytic combinatorics.

Contribution

It provides the first full asymptotic expansion for the enumeration of such graphs, improving understanding of their combinatorial structure.

Findings

Complete asymptotic expansion derived

Advances previous first-term results

Uses analytic combinatorics techniques

Abstract

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.