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

#### Abstract

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)).

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)).

#### Keywords

Total graph, generalization of total graphs

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