isomorphisms on annihilator graph of modules

Let r be a commutative ring with identity, and m be an r-module. in [1], we focus on annihilator graph of modules, ag(m), the sett(m)* is a non-zero torsion elements of m —would be the vertices of annihilator graph of modules, and x, y ∈ t(m)* were adjacent if and only if annr ([x m]y)≠ annr (x) ∪ annr (y) or annr ([y : m]x) ≠ annr (x) ∪ annr (y). we investigate the structure, the diameter, and the girth of annihilator graph of r-modules [1]. ghalandarzadeh and malakooti rad in [12] proved that for torsion graph of an r-module m is γ(m) whose vertices are nonzero torsion elements of m, and two distinct vertices x and y are adjacent if and only if [x : m][y : m]m = {0???? }, if s = r(m), then γ(m) and γ(???? −1m) are isomorphic for a multiplication r-module m. the purpose of this paper is to study the connection between the ag(m) and ag(???? −1m). we show that if s = r(m), then ag(m) and ag(???? −1m) are isomorphic for a multiplication r-module m.