On the graph \(G_P(R)\) over commutative ring R

Abstract

Let R be a commutative ring with identity 1. Then the graph of R, denoted by \(G_P(R)\) which is defined as the vertices are the elements of R and any two distinct elements a and b are adjacent if and only if the corresponding principal ideals aR and bR satisfy the condition: \((aR)(bR)=aR\bigcap bR\). In this paper, we characterize the class of finite commutative rings with 1 for which the graph \(G_P(R)\) is complete. Here we are able to show that the graph \(G_P(R)\) is a line graph of some graph G if and only if \(G_P(R)\) is complete. For \(n=p_1^{r_1}p_2^{r_2}\ldots p_{k}^{r_k}\), we show that chromatic number of \(G_P(\mathbb {Z}_n)\) is equal to the sum of the number of regular elements in \(\mathbb {Z}_n\) and the number of integers i such that \({r_{i}}>1\). Moreover, we characterize those n for which the graph \(G_P(\mathbb {Z}_n)\) is end-regular.