on annihilator graph of a finite commutative ring

‎the annihilator graph $ag(r)$ of a commutative ring $r$ is a simple undirected graph with the vertex set $z(r)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎in this paper we give the sufficient condition for a graph $ag(r)$ to be complete‎. ‎we characterize rings for which $ag(r)$ is a regular graph‎, ‎we show that $gamma (ag(r))in {1,2}$ and we also characterize the rings for which $ag(r)$ has a cut vertex‎. ‎finally we find the clique number of a finite reduced ring and characterize the rings for which $ag(r)$ is a planar graph‎.