Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say

, such that when M = R,

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

in the present article. We show that

is connected with

. We also show that for a reduced module M with

,

if and only if

is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple,

are adjacent if and only if

. Among other things, it is also observed that

if and only if M is uniform, ann(M) is a radical ideal, and

, if and only if ann(M) is prime and

.