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 2 M, the vertices x; y are adjacent if and only if x + y 2 MH(N) where MH(N) = fm 2 M : rm 2 N for some r 2 Hg and 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)).