The annihilating-submodule graph of modules over commutative rings

Let M be a module over a commutative ring R. The annihilating-submodule graph of M, denoted by AG(M), is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that NK = (0), where NK, the product of N and K, is denoted by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We obtain useful characterizations for thosemodules M for which either AG(M) is a complete graph or every vertex of AG(M) is a prime submodule of M. We prove that if AG(M) is a tree, then either AG(M) is a star graph or a path of order 4 and in the latter case M ∼ = F ×S, where F is a simple module and S is a module with a unique non-trivial submodule. Moreover, we prove that for every cyclic module M, cl(AG(M)) ≥|Min(M)|.