Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by Γ(M), is a graph whose vertices are the non-zero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = 0. In this paper, we investigate the relationship between the diameters of Γ(M) and Γ(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a Bézout ring R the diameter of Γ(M) and Γ(R) is equal, where M T(M). Also, we prove that, for a faithful multiplication R-module M with |M| 4, Γ(M) is a complete graph if and only if Γ(R) is a complete graph.
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