The generating graph of infinite abelian groups

For a group G, let Γ(G) denote the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Let Γ∗(G) be the subgraph of Γ(G) that is induced by all the vertices of Γ(G) that are not isolated. We prove that if G is a 2-generated noncyclic abelian group, then Γ∗(G) is connected. Moreover, diam(Γ∗(G)) = 2 if the torsion subgroup of G is nontrivial and diam(Γ∗(G)) = ∞ otherwise. If F is the free group of rank 2, then Γ∗(F) is connected and we deduce from diam(Γ∗(Z × Z)) = ∞ that diam(Γ∗(F)) =