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-zeroelements 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 undirectedsubmodule based graph ??fn(m) for each submodule n of m withvertices zfn(m) = {x 2 m n | xf(y) 2 n or yf(x) 2 n for some y 2m n}, in which non-zero elements x and y are adjacent provided thatxf(y) 2 n or yf(x) 2 n. we observe that over a commutative ring r,??fn(m) is connected and diam(??fn(m)) 6 3. also we get some resultsabout clique number and connectivity number of ??fn(m).