chromatic number of the perpendicular graphs of modules

‎let $r$ be a ring and $m$ be an $r$-module‎. ‎two modules $a$ and‎ ‎$b$ are called orthogonal‎, ‎written $aperp b$‎, ‎if they do not have‎ ‎non-zero isomorphic submodules‎. ‎we consider an associated graph $gamma_{ot}(m)$ to $m$‎ with vertices $mathcal{m}_{perp}={(0)eq alneq m;|; exists (0)eq blneq m ; mbox{such that}; aperp b}$‎, ‎ and for distinct $a,bin‎mathcal{m}_{perp}$‎, ‎the vertices $a$ and $b$ are adjacent if and only if‎‎$aperp b$‎. ‎the main object of this article is to study the‎ ‎interplay of module-theoretic properties of $m$ with‎ graph-theoretic properties of $gamma_{ot}(m)$‎. ‎we study the clique number and chromatic number of $gamma_{ot}(m)$‎. among other results‎, ‎we study when $omega(gamma_{ot}(m)) < infty $‎, conclude that $chi(gamma_{ot}(m)) < infty $‎. ‎also it is shown that for semi-artinian module $m$‎, ‎$omega(gamma_{ot}(m))=chi(gamma_{ot}(m))$‎.