On the diameters of McKay graphs for finite simple groups

Let Gbe a finite group, andαa nontrivial character of G. The McKay graph M(G,α) has the irreducible characters of Gas vertices, with an edge from χ1 to χ2 if χ2 is a constituent of αχ1. We study the diameters of McKay graphs for simple groups G of Lie type. We show that for anyα, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for G= PSLn(q) or PSUn(q