The Zariski topology-graph of modules over commutative rings II

Let $M$ be a module over a commutative ring $R$. In this paper, we continue our study about the Zariski topology-graph $G({\tau }_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283–3296). For a non-empty subset $T$ of $Spec(M)$, we obtain useful characterizations for those modules $M$ for which $G({\tau }_T)$ is a bipartite graph. Also, we prove that if $G({\tau }_T)$ is a tree, then $G({\tau }_T)$ is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $\chi (G({\tau }_T))$ and $\omega (G({\tau }_T))$.