The non-isolated vertices in the generating graph of a direct powers of simple groups

For a finite group G let I"(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Many deep results on the generation of the finite simple groups G can be equivalently stated as theorems that ensure that I"(G) is a rich graph, with several good properties. In this paper we want to consider I"(G (delta) ) where G is a finite non-abelian simple group and G (delta) is the largest 2-generated power of G, with the aim to investigate whether the good generation properties of G still affect the behaviour of I"(G (delta) ). In particular we prove that the graph obtained from I"(G (delta) ) by removing the isolated vertices is 1-arc transitive and connected and we investigate the diameter of this graph. Moreover, some intriguing open questions will be introduced and their solutions will be exemplified for .