Let G be a finite group, andαa nontrivial character of G. The McKay graph M (G,α) has the irreducible characters of Gas vertices, with an edge fromχ1toχ2ifχ2is a constituent ofαχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G=An, we prove a conjecture made in [20]: there is an absolute constant C such that diam M (G,α)≤ C log | G| log α (1)for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a productχ1χ2···χlof irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents
