Mckay graphs for alternating and classical groups

Let G be a finite group, and α a nontrivial character of G. The McKay graphM(G, α) has the irreducible characters of G as vertices, with an edge from χ1 to χ2 if χ2 is a constituent of αχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G = An, we prove a conjecture made in another work by the authors: There is an absolute constant C such that diamM(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 product χ1χ2 χl of irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents.