A module theoretic approach to‎ ‎zero-divisor graph with respect to (first) dual

Let M be an R-module and 0≠finM∗=Hom(M,R). We associate an undirected graph gf to M in which non-zero elements x and y of M are adjacent provided that xf(y)=0 or yf(x)=0. We observe that over a commutative ring R, gf is connected and diam(gf)≤3. Moreover, if Γ(M) contains a cycle, then gr(gf)≤4. Furthermore if |gf|geq1, then gf is finite if and only if M is finite. Also if gf=emptyset, then f is monomorphism (the converse is true if R is a domain). If M is either a free module with rank(M)≥2 or a non-finitely generated projective module there exists finM∗ with rad(gf)=1 and diam(gf)≤2. We prove that for a domain R the chromatic number and the clique number of gf are equal.