The diameter of random graphs

Abstract

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that ifd=d(n)⩾3d = d(n) \geqslant 3andm=m(n)m = m(n)satisfy(log⁡n)/d−3log⁡log⁡n→∞(\log n)/d - 3\,\log \log n \to \infty,2d−1md/nd+1−log⁡n→∞{2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \inftyanddd−2md−1/nd−log⁡n→−∞{d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \inftythen almost every graph withnnlabelled vertices andmmedges has diameterdd.