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 Γ(S^n) be the generating graph of S^n and let Γ_n (S) be the graph obtained from Γ(S^n) by removing all isolated vertices. A recent result of Crestani and Lucchini states that Γ_n(S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(Γ_1(S))=2, and we define Δ(S) to be the maximal n such that diam(Γ_n (S))=2. We prove that Δ(S)≥2 for all S, which is best possible since Δ(A_5)=2, and we show that Δ(S) tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups.