Asymptotic enumeration of strongly connected digraphs by vertices and edges

TL;DR

This paper derives an asymptotic formula for counting strongly connected directed graphs with a given number of vertices and edges, bridging previous results for sparse and dense regimes.

Contribution

It provides a new asymptotic enumeration formula for strongly connected digraphs in the intermediate range of edges, extending prior work.

Findings

Asymptotic formula valid for m-n→∞ with m=O(n log n)

Bridges gap between sparse and dense regimes

Enhances understanding of connectivity thresholds in random digraphs

Abstract

We derive an asymptotic formula for the number of strongly connected digraphs with $n$ vertices and $m$ arcs (directed edges), valid for $m-n\to\infty$ as $n\to \infty$ provided $m=O(n\log n)$. This fills the gap between Wright's results which apply to $m=n+O(1)$, and the long-known threshold for $m$, above which a random digraph with $n$ vertices and $m$ arcs is likely to be strongly connected.