Non-nilpotent element graph of a module over a commutative ring

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Gauhati University, Guwahati-14, Assam, India

Abstract

Let $R$ be a commutative ring with non-zero unity and $M$ be a unitary $R$-module. Let $Nil(M)$ be the set of all the nilpotent elements of $M$ and $Non(M)=M-Nil(M)$ be the set of all non-nilpotent elements of $M$. The non-nilpotent element graph of $M$ over $R$ is an undirected simple graph $G_{NN}(M)$ with $Non(M)$ as vertex set and any two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Nil(M)$. In this paper, we study the basic properties of the graph $G_{NN}(M)$. We also study the diameter and girth of $G_{NN}(M)$. Further, we determine the domination number and the bondage number of $G_{NN}(M)$. We establish a relation between the diameter and domination number of $G_{NN}(M)$. We also establish a relation between the girth and bondage number of $G_{NN}(M)$.

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References

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Algebraic Structures and Their Applications

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