The Asymptotic Number of Unlabelled Regular Graphs

Over ten years ago Wright [4] proved a fundamental theorem in the theory of random graphs. He showed that if M = M(n) is such that almost no labelled graph of order n and size M has two isolated vertices or two vertices of degree n — 1, then the number of labelled graphs of order n and size M divided by the number of unlabelled graphs of order n and size M is asymptotic to n\. The result is best possible, since if the above ratio is asymptotic to n\ then almost no labelled graph of order n and size M has a non-trivial automorphism. The aim of this paper is to prove the analogue of Wright's theorem for regular graphs. Random regular graphs have not been studied for long. The main reason for this is that until recently there was no asymptotic formula for the number of labelled regular graphs: such a formula was found by Bender and Canfield [1]. Even more recently, in [2] a model was given for the set of labelled regular graphs which makes the study of random regular graphs fairly accessible. In particular, a result in [3] implies that, for r ^ 3, almost every labelled r-regular graph has only the identity as its automorphism. Let r ^ 3 be fixed and let n -* oo in such a way that rn = 2m is even. Denote by S£r = S£KtT the set of r-regular graphs with vertex set V = { l ,2 , . . . ,n} . Write °llr = °Unr for the set of unlabelled r-regular graphs of order n. Put Lr = \££?r\ and Ur = \%\. We know from [1] that ^ (r2_1)/4(2m)! Lr 2m\ •