A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs

Abstract

Let Δ and n be natural numbers such that Δn = 2m is even and Δ ⩽ (2 log n )1/2 - 1. Then as n →, the number of labelled Δ-regular graphs on n vertices is asymptotic to e −λ−λ2 (2m)! m!2m (Δ!)m where λ = (Δ -1)/2. As a consequence of the method we determine the asymptotic distribution of the number of short cycles in graphs with a given degree sequence, and give analogous formulae for hypergraphs. © 1980, Academic Press Inc. (London) Limited. All rights reserved.

Publication Title

European Journal of Combinatorics

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