ON THE EXTENDED TOTAL GRAPH OF MODULES OVER COMMUTATIVE RINGS

Authors : F Esmaeili Khalil SARAEİ, E NAVİDİNİA

Pages : 77-86

Doi:10.24330/ieja.504118

View : 13 | Download : 12

Publication Date : 2019-01-08

Article Type : Research Paper

Abstract : Let $M$ be a module over a commutative ring $R$ and $U$ a nonempty proper subset of $M$. In this paper, the extended total graph, denoted by $ET_{U}insert ignore into journalissuearticles values(M);$, is presented, where  $U$ is a multiplicative-prime subset of $M$. It is the graph with all elements of $M$ as vertices, and for distinct $m,n\in M$, the vertices $m$ and $n$ are adjacent if and only if $rm+sn\in U$ for some $r,s\in R\setminus insert ignore into journalissuearticles values(U:M);$. We also study the two insert ignore into journalissuearticles values(induced); subgraphs $ET_{U}insert ignore into journalissuearticles values(U);$ and $ET_{U}insert ignore into journalissuearticles values(M\setminus U);$, with vertices $U$ and $M\setminus U$, respectively. Among other things, the diameter and the girth of $ET_{U}insert ignore into journalissuearticles values(M);$ are also studied.
Keywords : Total graph, prime submodule, multiplicative prime subset