Asymptotic enumeration by degree sequence of graphs with degreeso(n ^1/2)

Abstract

We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|^1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|^1/4). In particular, ifk=o(n ^1/2), the number of regular graphs of degreek and ordern is asymptotically

$$\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).$$

Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.