### Abstract:

Let R be a commutative ring and Z(R)
∗ be its
set of non-zero zero-divisors. The annihilator graph of a commu-
tative ring R is the simple undirected graph AG(R) with vertices
Z(R)
∗
, and two distinct vertices x and y are adjacent if and only
if ann(xy) 6= ann(x) ∪ 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 AG(R) has genus less or equal to one.