Line graph of extensions of the zero-divisor graph in commutative rings

Abstract

UDC 512.6

We consider a finite commutative ring with unity denoted by $\mathscr{P}.$ Within this framework, the essential graph of $\mathscr{P}$ is represented as $E{G}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann(xy)$ is an essential ideal of $\mathscr{P}.$ At the same time, the weakly zero-divisor graph of $\mathscr{P}$ is denoted by $\text{W}{\Gamma}(\mathscr{P})$ with $Z(\mathscr{P})^* = Z(\mathscr{P})\setminus\{0\}$ as the vertex set and an edge is defined between two distinct vertices $u$ and $v$ if and only if there exist $r \in ann(u)^*$ and $s\in ann(v)^*$ such that $rs=0$, where $ann(u) = \{v \in \mathscr{P}\colon uv = 0\}$ for $u \in \mathscr{P}.$ In our research, we deal with the conditions under which both $E{G}(\mathscr{P})$ and $\text{W}{\Gamma}(\mathscr{P})$ can be classified as line graphs. Furthermore, we explore the scenarios in which these graphs are the complements of line graphs.