The generating graph of the abelian groups

For a group $G,$ let $Gamma(G)$ 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 $G$. Moreover let $Gamma^*(G)$ be the subgraph of $Gamma(G)$ that is induced by all the vertices of $Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Gamma^*(G)$ is connected. Moreover $mathrm{diam}(Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $mathrm{diam}(Gamma^*(G))=infty$ otherwise. If $F$ is the free group of rank 2, then $Gamma^*(F)$ is connected and we deduce from $mathrm{diam}(Gamma^*(mathbb{Z}times mathbb{Z}))=infty$ that $mathrm{diam}(Gamma^*(F))=infty.$