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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