The diameter of sparse random graphs

We consider the diameter of a random graph G(n, p) for various ranges of p close to the phase transition point for connectivity. For a disconnected gr aph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of ra ndom graph G(n, p) is close to logn/log (np) if np --> infinity. Moreover i f np/log n = c > 8, then the diameter of C(n, p) is concentrated on two val ues. In general, if np/log n = C > C-0, the diameter is concentrated on at most 2 [1/c(0)] + 4 values. We also proved that the diameter of G(n, p) is almost surely equal to the diameter of its giant component if np > 3.6. (C) 2001 Academic Press.