Asymptotic Enumeration by Degree Sequence of Graphs of High Degree

We consider the estimation of the number of labelled simple graphs with degree sequence d1, d2, . . . , dn by using an n-dimensional Cauchy integral. For suffiently small and any c > 2 3 , an asymptotic formula is obtained when |di − d| < n for all i and d = d(n) satisfies min{d, n − d − 1} ≥ cn/ log n as n → ∞. These conditions include the degree sequences of almost all graphs, so our result gives as a corollary the asymptotic joint distribution function of the degrees of a random graph. We also give evidence for a formula conjectured to be valid for all d(n).