The generating graph of the abelian groups

Abstract: 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 $\mathrm{diam}(\Gamma^*(F))=\infty.$