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(n1/2), the number of regular graphs of degreek and ordern is asymptotically
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.