Abstract: | For each of the two models of a sparse random graph on n vertices, G(n, # of edges = cn/2) and G(n, Prob (edge) = c/n) define tn(k) as the total number of tree components of size k (1 ≤ k ≤ n). the random sequence {tn(k) - nh(k)]n?1/2} is shown to be Gaussian in the limit n →∞, with h(k) = kk?2ck?1e?kc/k! and covariance function being dependent upon the model. This general result implies, in particular, that, for c> 1, the size of the giant component is asymptotically Gaussian, with mean nθ(c) and variance n(1 ? T)?2(1 ? 2Tθ)θ(1 ? θ) for the first model and n(1 ? T)?2θ(1 ? θ) for the second model. Here Te?T = ce?c, T<1, and θ = 1 ? T/c. A close technique allows us to prove that, for c < 1, the independence number of G(n, p = c/n) is asymptotically Gaussian with mean nc?1(β + β2/2) and variance nc?1(β + β2/2) ?c?2(c + 1)β2], where βeβ = c. It is also proven that almost surely the giant component consists of a giant two-connected core of size about n(1 ? T)β and a “mantle” of trees, and possibly few small unicyclic graphs, each sprouting from its own vertex of the core. |