The Submodule-Based Zero-Divisor Graph with Respect to Some Homomorphism

Let M be an R-module and 0 6= f 2 M = Hom(M,R). The graph ??f (M) is a graph with vertices Zf (M) = {x 2 M \ {0} | xf(y) = 0 or yf(x) = 0 for some non-zero y 2 M}, in which non-zero elements x and y are adjacent provided that xf(y) = 0 or yf(x) = 0, which introduced and studied in [3]. In this paper we associate an undirected submodule based graph ??f N(M) for each submodule N of M with vertices Zf N(M) = {x 2 M \ N | xf(y) 2 N or yf(x) 2 N for some y 2 M \N}, in which non-zero elements x and y are adjacent provided that xf(y) 2 N or yf(x) 2 N. We observe that over a commutative ring R, ??f N(M) is connected and diam(??f N(M)) 6 3. Also we get some results about clique number and connectivity number of ??f N(M).