Some larger trivalent graphs having small diameters

This paper concerns an improvement of a result of Babai, Kantor and Lubotzky. It was shown that there is a constant C such that every non-Abelian finite simple group G has a set S of seven generators for which . Here, S was a carefully chosen generating set for G, and denotes the diameter of the corresponding undirected Cayley graph. This bound is best possible, since a simple count (the ‘Moore bound’) shows that .