On the diameter of Cayley graphs of classical groups with generating sets containing a transvection

A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set for G, then the diameter of the Cayley graph Cay(G, X) is bounded by log ∣G∣c for some universal constant c. In this paper, we prove such a bound for Cay(G, X) for G = PSL(n, q), PSp(n, q) or PSU(n, q) where q is odd, under the assumptions that X contains a transvection and q ≠ 9 or 81.