Maryam Shirali - Department of Mathematics, University of Yasouj, Yasouj, Iran.
Saeid Safaeeyan - Department of Mathematics, University of Yasouj, Yasouj, Iran.

‎In this paper we continue our study of perpendicular graph of modules‎, ‎that was introduced in \cite{Hokkaido}‎. ‎Let R be a ring and M be an R-module‎. ‎Two modules A and‎ ‎B are called orthogonal‎, ‎written A\perp B‎, ‎if they do not have‎ ‎non-zero isomorphic submodules‎. ‎We associate a graph \Gamma_{\bot}(M) to M‎ ‎with vertices‎ ‎\mathcal{M}_{\perp}=\{(۰)\neq A\leq M\;|\; \exists (۰)\neq B\leq M \; \mbox{such that}\; A\perp B\}‎, ‎and for distinct A,B\in‎ ‎\mathcal{M}_{\perp}‎, ‎the vertices A and B are adjacent if and only if‎ ‎A\perp B‎. ‎The main object of this article is to study the‎ ‎interplay of module-theoretic properties of M with‎ ‎graph-theoretic properties of \Gamma_{\bot}(M)‎. ‎We study the clique number and chromatic number of \Gamma_{\bot}( M)‎. ‎We prove that if \omega(\Gamma_{\bot}( M)) < \infty and M has a simple submodule‎, ‎then \chi(\Gamma_{\bot}(M)) < \infty ‎. ‎Among other results‎, ‎it is shown that for a semi-simple module M‎, ‎\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))‎.

chromatic number‎, ‎clique number‎, ‎finite ‎graph, atomic module, semi-simple module‎‎