Brownian approximation to counting graphs

TL;DR

This paper establishes a connection between the enumeration of connected graphs with specific edge counts and the moments of the Brownian excursion's area, extending previous asymptotic results.

Contribution

It provides a new elementary proof linking graph enumeration to Brownian excursion moments for sequences where k_n=o(∛n), expanding prior results.

Findings

Asymptotic equivalence of graph counts and Brownian excursion moments for k_n=o(∛n)

Use of strong embedding of empirical processes in Brownian bridges

Extension of previous asymptotic results to a broader range of k_n

Abstract

Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is asymptotically equal to the $k_n$th moment of the area under the standard Brownian excursion. These moments have been computed in the literature via independent methods. In this article we show why this is true for k_n=o(\sqrt[3]{n}) starting from an observation made by Joel Spencer. The elementary argument uses a result about strong embedding of the Uniform empirical process in the Brownian bridge proved by Komlos, Major, and Tusnady.