some results on annihilator graph

Beck-anderson-livingston zero-divisor graph of r is the (simple) graph γ(r) with vertices z(r) n f0g, anddistinct vertices x and y are adjacent if and only if xy = 0. in 2014, badawi introduced the annihilatorgraph of r. the annihilator graph of r is the (undirected) graph ag(r) with vertices z(r)∗ = z(r) n f0g,and two distinct vertices x and y are adjacent if and only if annr(xy) 6= annr(x) [ annr(y) (recall if m inr, then annr(m) = fy 2 r j ym = 0g). it follows that each edge (path) of the classical zero-divisor of r isan edge (path) of ag(r). in this talk, we state a number of results about some graphs as in [1], [2], [3] and[4]. among many results, we state the following.