On the diameters of McKay graphs for finite simple groups

Let G be a finite group, and α a nontrivial character of G . The McKay graph ℳ ( G , α ) has the irreducible characters of G as vertices, with an edge from χ 1 to χ 2 if χ 2 is a constituent of αχ 1 . We study the diameters of McKay graphs for simple groups G of Lie type. We show that for any α , the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for G = PSL n ( q ) or PSU n ( q ).