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

Abstract

Let d(n, q) be the number of labeled graphs with n vertices, q ≤ N = (n2) edges, and no isolated vertices. Let x = q/n and k = 2q - n. We determine functions wk ∼ 1, a(x), and φ(x) such that d(n, q) ∼ wk(Nq)enφ(x)+a(x) uniformly for all n and q > n/2. © 1997 Academic Press.