Let M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph G(τ 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(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 Zariski topology-graphs and investigate the interplay between χ(G(τ T )) and ω(G(τ T )).2010 Mathematics Subject Classification. 13C13, 13C99, 05C75.

Let M be a module over a commutative ring and let Spec(M ) be the collection of all prime submodules of M . We topologize Spec(M ) with quasi-Zariski topology and, for a subset T of Spec(M ), we introduce a new graph G(τ * T ), called the quasi-Zariski topology-graph. It helps us to study algebraic (respectively, topological) properties of M (respectively, Spec(M )) by using graph theoretical tools. Also, we study the annihilating-submodule graph and investigate the relation between these two graphs. 2010 AMS Mathematics subject classification. Primary 13C13, 13C99.

Let R be a commutative ring and let M be a finitely generated R-module. Let's denote the cozero-divisor graph of R by K .R/. In this paper, we introduce a certain subgraph K R .M / of K .R/, called cozero-divisor graph relative to M , and obtain some related results.

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