On the genus of the annihilator graph of a commutative ring

Let $R$ be a commutative ring and $Z(R)^*$ be its set of non-zero zero-divisors. The annihilator graph of a commutative ring $R$ is the simple undirected graph $\operatorname{AG}(R)$ with vertices $Z(R)^*$, and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}(xy)\neq \operatorname{ann}(x)\cup \operatorname{ann}(y)$. The notion of annihilator graph has been introduced and studied by A. Badawi [7]. In this paper, we determine isomorphism classes of finite commutative rings with identity whose $\operatorname{AG}(R)$ has genus less or equal to one.