Counting strongly-connected, moderately sparse directed graphs

DOI10.1002/RSA.20433zbMath1270.05059arXiv1005.0327OpenAlexW2120610734MaRDI QIDQ2844082

Boris G. Pittel

Publication date: 27 August 2013

Published in: Unnamed Author (Search for Journal in Brave)

Abstract: A sharp asymptotic formula for the number of strongly connected digraphs on $n$ labelled vertices with $m$ arcs, under a condition $m-n oinfty$, $m=O(n)$, is obtained; this solves a problem posed by Wright back in $1977$. Our formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on $n$ vertices with $m$ edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Nick Wormald we rederived the BCM formula via counting two-connected graphs among the graphs of minimum degree $2$, at least. In this paper, using a similar embedding for directed graphs, we find an asymptotic formula, which includes an explicit error term, for the fraction of strongly-connected digraphs with parameters $m$ and $n$ among all such digraphs with positive in/out-degrees.

Full work available at URL: https://arxiv.org/abs/1005.0327

zbMATH Keywords

asymptoticsenumerationdirected graphsstrongly connected

Mathematics Subject Classification ID

Enumeration in graph theory (05C30) Directed graphs (digraphs), tournaments (05C20) Connectivity (05C40)

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Related Items (3)

The critical window in random digraphs ⋮ Birth of a giant \((k_{1},k_{2})\)-core in the random digraph ⋮ Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

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