The central vertices and radius of the regular graph of ideals

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.