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

Let M be an R-module and 0 6= f ∈ M∗ = Hom(M, R). The graph Γf (M) is a graph with vertices Z f (M) = {x ∈ M \ {0} | xf(y) = 0 or yf(x) = 0 for some non-zero y ∈ 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 ΓfN (M) for each submodule N of M with vertices Z N (M) = {x ∈ M \ N | xf(y) ∈ N or yf(x) ∈ N for some y ∈ M \N}, in which non-zero elements x and y are adjacent provided that xf(y) ∈ N or yf(x) ∈ N . We observe that over a commutative ring R, ΓfN (M) is connected and diam(Γ f N (M)) 6 3. Also we get some results about clique number and connectivity number of ΓfN (M) AMS Subject Classification: 05C25; 05C38; 05C40; 16D10; 16D4