A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs
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.
European Journal of Combinatorics
Bollobás, B. (1980). A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs. European Journal of Combinatorics, 1 (4), 311-316. https://doi.org/10.1016/S0195-6698(80)80030-8