Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(τ∗T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1–19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(τ∗T) is a bipartite graph. Also, we prove that if G(τ ∗ T) is a tree, then G(τ∗T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between χ(G(τ∗T)) and ω(G(τ ∗ T))