On the Planarity of Extended Zero-Divisor Graphs of Local Rings

Abstract

Let $R$ be a commutative ring with unity $1\ne 0$ . Recently Bennis et al. defined the concept of extended zero-divisor graph $\overline{\mathrm{\Gamma }}\left(R\right)$ by considering the vertex set $V\left(\overline{\mathrm{\Gamma }}\left(R\right)\right)=Z^{\ast }\left(R\right)$ and any two vertices $x$ and $y$ are adjacent if there exist positive integers $m$ and $n$ , such that $x^m{y}^n=0$ with $x^m\ne 0$ and $y^n\ne 0$ . The main objective of this article is to check the planar property of extended zero-divisor graphs. Also, a complete list of local rings up to order 27 with planar extended zero-divisor graphs has been collected.