Abstract
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 $Z_R(M) setminus {rm Ann}_R(M)$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${rm Ann}_M(xy)neq {rm Ann}_M(x) cup {rm Ann}_M(y)$. 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.
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