further studies of the perpendicular graphs of modules

In this paper we continue our study of perpendicular graph of modules, that was introduced in [7]. let r be a ring and m be an r-module. two modules a and b are called orthogonal, written a ⊥ b, if they do not have non-zero isomorphic submodules. we associate a graph γ⊥(m) to m with vertices m⊥ = {(0) ̸= a ≤ m | ∃(0) ̸= b ≤ m such that a ⊥ b}, and for distinct a,b ∈m⊥, the vertices a and b are adjacent if and only if a ⊥ b. the main object of this article is to study the interplay of module-theoretic properties of m with graph-theoretic properties of γ⊥(m). we study the clique number and chromatic number of γ⊥(m). we prove that if ω(γ⊥(m)) < ∞ and m has a simple submodule, then χ(γ⊥(m)) < ∞. among other results, it is shown that for a semi-simple module m, ω(γ⊥(rm)) = χ(γ⊥(rm)).