The Center and Radius of the Regular Graph of Ideals

TL;DR

This paper investigates the regular graph of ideals in a commutative ring, establishing that its radius is always 3 and identifying its central vertices, thus advancing understanding of its structural properties.

Contribution

It proves that the radius of the regular graph of ideals is always 3 and characterizes the central vertices, providing new insights into its structure.

Findings

Radius of the graph is always 3

Central vertices are characterized

Structural properties of the graph are elucidated

Abstract

The regular graph of ideals of the commutative ring $R$, denoted by ${\Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $\Gamma_{reg}(R)$ equals $3$. The central vertices of $\Gamma_{reg}(R)$ are determined, too.