Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${Gamma}(M)$, such that when M = R, ${Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${Gamma}(M)$ in the present article. We show that ${Gamma}(M)$ is connected with $diam({Gamma}(M)){leq}3$. We also show that for a reduced module M with $Z(M)^*{ eq}M{ackslash}{0}$, $gr({Gamma}(M))={infty}$ if and only if ${Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{in}M{ackslash}{0}$ are adjacent if and only if $xR{cap}yR=(0)$. Among other things, it is also observed that ${Gamma}(M)={emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{ eq}M{ackslash}{0}$, if and only if ann(M) is prime and $Z(M)^*{ eq}M{ackslash}{0}$.