A Generalization of Total Graphs of Modules over Commutative Rings under Multiplicatively Closed Subsets

Let r be a commutative ring and m be an r-module with a proper submodule n. a generalization of total graphs, denoted by t(γn h(m)), is introduced and investigated. it is the (undirected) graph with all elements of m as vertices and for distinct x, y ∈ m, the vertices x, y are adjacent if and only if x+y ∈ mh(n) where mh(n) = {m ∈ m : rm ∈ n for some r ∈ h}, where h is a multiplicatively closed subset of r. in this paper, in addition to studying some algebraic properties of mh(n), we investigate some graph theoretic properties of two essential subgraphs of t(γn h(m)).