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 if d = d ( n ) ⩾ 3 d = d(n) \geqslant 3 and m = m ( n ) m = m(n) satisfy ( log ⁡ n ) / d − 3 log ⁡ log ⁡ n → ∞ (\log n)/d - 3\,\log \log n \to \infty , 2 d − 1 m d / n d + 1 − log ⁡ n → ∞ {2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \infty and d d − 2 m d − 1 / n d − log ⁡ n → − ∞ {d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \infty then almost every graph with n n labelled vertices and m m edges has diameter d d .