the annihilator graph for moduled over commutative rings

Let r be a commutative ring and m be an r-module. the annihilator graph of m, denoted by ag(m) is a simple undirected graph associated to m whose the set of vertices is zr(m) annr(m) and two distinct vertices x and y are adjacent if and only if annm(xy) ≠ annm(x) ∪ annm(y). in this paper, we study the diameter and the girth of ag(m) and we characterize all modules whose annihilator graph is complete. furthermore, we look for the relationship between the annihilator graph of m and its zero-divisor graph.