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.