Diameter and girth of Torsion Graph

Let R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the corresponding torsion graph, Γ_R(M), with two distinct vertices x, y ∈ T(M)* forming an edge if Ann(x) ∩ Ann(y) ≠ 0. In this paper we study the case where the graph Γ_R(M) is connected with diam(Γ_R(M)) ≤ 3 and we investigate the relationship between the diameters of Γ_R(M) and Γ_R(R). Also we study girth of Γ_R(M), it is shown that if Γ_R(M) contains a cycle, then gr(Γ_R(M)) = 3.