On the generating graph of direct powers of a simple group

Let S be a nonabelian finite simple group and let n be an integer such that the direct product S (n) is 2-generated. Let I(S ^n ) be the generating graph of S^n and let I'_n(S) be the graph obtained from I(S^n ) by removing all isolated vertices. A recent result of Crestani and Lucchini states that I-_n(S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(I_1(S))=2, and we define Delta(S) to be the maximal n such that diam(I_n (S))=2. We prove that Delta(S) is at least 2 for all S, which is best possible since Delta(Alt(5))=2, and we show that Delta(S) tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups.