On line upper ideal relation graphs of rings

Abstract

The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the principal ideals $(x)$ and $(y)$ are contained in the principal ideal $(z)$ for some non-unit element $z\in R$. This manuscript characterizes all the Artinian rings $R$ such that the graph $\Gamma_{U}(R)$ is a line graph. Moreover, all the Artinian rings $R$ for which $\Gamma_{U}(R)$ is the complement of a line graph have been described.