On the line graph structure of the cozero-divisor graph of a commutative ring

Abstract

Assume that R is a commutative ring with non-zero identity and $W^*(R)$ is the set of all non-zero non-unit elements of R. Also, for $x\in R$, the ideal which is generated by x, is denoted by Rx. The cozero-divisor graph of R, which is denoted by $\Gamma '(R)$, is a graph with $W^*(R)$ as the vertex-set, and two distinct vertices x and y are adjacent in $W^*(R)$ if and only if $x\notin Ry$ and $y\notin Rx$. In this paper, we completely determine all finite commutative rings R such that $\Gamma '(R)$ is a line graph. We also characterize all finite commutative rings R such that $\Gamma '(R)$ is isomorphic to its line graph.