Complement of the co-intersection graph of submodules of a module

Let [Formula: see text] be a ring with identity and [Formula: see text] be a unitary left [Formula: see text]-module. The complement of the co-intersection graph of submodules of a module [Formula: see text], denoted by [Formula: see text], is defined to be a undirected simple graph whose vertices are in one-to-one correspondence with all nontrivial submodules of [Formula: see text] and two distinct vertices are adjacent if and only if the sum of corresponding submodules of [Formula: see text] is equal to [Formula: see text]. In this paper, we consider the complement of the co-intersection graph of submodules of a module. We study the connectivity, the diameter and the girth of [Formula: see text]. We prove that, if [Formula: see text] and [Formula: see text], then [Formula: see text]. We show that, if [Formula: see text] is a finitely generated [Formula: see text]-module and [Formula: see text] contains a cycle such that [Formula: see text], then the core of [Formula: see text] is a union of [Formula: see text]-cycles and [Formula: see text]-cycles and also every vertex of [Formula: see text] is either an end vertex or a vertex of the core. We determine the structure of modules whose associated graphs are bipartite, complete bipartite, planar and weakly perfect.