On the graph of modules over commutative rings II

Habibollah Ansari-Toroghy, Shokoofeh Habibi, Masoomeh Hezarjaribi


Let $M$ be a module over a commutative ring $R$. In this paper, we continue our study about the quasi-Zariski topology-graph
$G(\tau^*_T)$ 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 $G(\tau^*_T)$ 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
quasi-Zariski topology-graphs and investigate the interplay
between $\chi(G(\tau^*_T))$ and $\omega(G(\tau^*_T))$.


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