The asymptotic number of labeled graphs with n vertices, q edges, and no isolated vertices

摘要

Let d(n, q) be the number of labeled graphs with n vertices, q less than or equal to N = ((n)(2)) edges. and no isolated vertices. Let x = q and k = 2q - n. We determine functions, w(k) similar to 1. a(x), and phi(x) such that d(n, q) similar to w(k)((N)(q)) e(n phi(x) + a(x)) uniformly for all n and q > n/2. (C) 1997 Academic Press.