The Asymptotic Number of Labeled Graphs withnVertices,qEdges, and No Isolated Vertices

Letd(n, q) be the number of labeled graphs withnvertices, q\le N={n \choose 2} edges, and no isolated vertices. Letx=q/nandk=2q−n. We determine functionswk∼1,a(x), andϕ(x) such that d(n,\ q)\sim w_k{N \choose q}\ e^{n\varphi (x)+a(x)} uniformly for allnandq>n/2.