The modified divisor graph of commutative ring

Abstract

Let \( R \) be a finite commutative ring with unity. We have introduced a divisor graph of the ring, denoted by \( D[R] \). It is an undirected simple graph with a vertex set \( V = R - \{0, 1\} \), and two distinct vertices \( u \) and \( v \) in \( V \) are adjacent if and only if there exists \( w \in R \) such that \( v = uw \) or \( u = vw \). Clearly, \( 0 \) and unit elements are adjacent to all elements of the ring. Thus, we adopt the idea of Anderson and Livingston and remove the zero and unit vertices from the vertex set. A new graph is called the modified divisor graph, noted by symbol \( D_0[R] \).In this study, we have focused on the structural properties of \( D_0[\mathbb{Z}_n] \). Moreover, we have determined the diameter, girth, clique number, vertex connectivity, and edge connectivity of \( D_0[\mathbb{Z}_n] \) for every natural number \( n \). The purpose of studying this graph is to find relationships between the ring theoretic properties of \( R \) and the graph theoretic properties of \( D_0[R] \).