Shahrood University of TechnologyJournal of Algebraic Systems2345-512812120240901NON-NILPOTENT GRAPH OF COMMUTATIVE RINGS149162284310.22044/jas.2023.12064.1620ENHussain Mohammed Imdadul HoqueDepartment of Mathematics, Gauhati University, Guwahati-781014, India.Helen KumariSaikiaDepartment of Mathematics, Gauhati University, Guwahati-781014, India.Jituparna GoswamiDepartment of Mathematics, Gauhati University, Guwahati-14, Assam, India.0000-0002-1786-752XDiksha PatwariDepartment of Mathematics, Gauhati University, Guwahati-781014, India.Journal Article20220704Let R be a commutative ring with unity. Let Nil(R) be the set of all nilpotent elements of R and Nil(R) = R \ Nil(R) be the set of all non-nilpotent elements of R. The non-nilpotent graph of R is a simple undirected graph GNN(R) with Nil(R) as vertex set and any two distinct vertices x and y are adjacent if and only if x+y ∈ Nil(R).<br />In this paper, we introduce and discuss the basic properties of the graph GNN(R). We also study the diameter and girth of GNN(R). Further, we determine the domination number and the bondage number of GNN(R). We establish a relation between diameter and domination number of GNN(R). We also establish a relation between girth and bondage number of GNN(R).https://jas.shahroodut.ac.ir/article_2843_3c09101e97af59b1db3801b6537792ce.pdf