center 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 a ⊥ b, if they do not have non-zero isomorphic submodules. we consider an associated 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 objectof this article is to study the interplay of module-theoretic properties of m with graph-theoretic properties of γ⊥(m). in this paper we study the center of perpendicular graph of modules.