the total graph of a module with respect to multiplicative-prime subsets

Let M be a module over a commutative ring R and U a nonempty proper subset of M. In this paper, a generalization of the total graph T(Γ(M)), denoted by T(Γ_U (M)) is presented, where U is a multiplicative prime subset of M. It is the graph with all elements of M as vertices, and for two distinct elements m, n ∈ M, the vertices m and n are adjacent if and only if m + n ∈ U. The main purpose of this paper is to extend the definitions and properties given in [1] and [10] to a more general case.