  Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The
McKay graph ${\mathcal M}(G,\alpha)$ has the irreducible characters of $G$ as
vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent
of $\alpha\chi_1$. We study the diameters of McKay graphs for simple groups
$G$. For $G$ a group of Lie type, we show that for any $\alpha$, the diameter
is bounded by a quadratic function of the rank, and obtain much stronger bounds
for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for
symmetric and alternating groups.
