University of Guilan Journal of Algebra and Related Topics 2345-3931 Articles in Press 2026 02 10 Planarity of the essential graph for modules 9435 10.22124/jart.2025.30608.1799 EN S. Azlesh Department of Mathematics, Imam Khomeini International University, Qazvin, Iran Sh. Payrovi Department of Mathematics, Imam Khomeini International University, Qazvin, Iran F. Soheilnia Department of Mathematics, Imam Khomeini International University, Qazvin, Iran Journal Article 2025 05 07 ‎Given a module $\mathcal{M}$ over a commutative ring $\mathcal{R}$‎, ‎we can construct a simple graph‎ ‎$EG\mathcal{(M)}$ with the vertex set $\mathcal{Z_R(M)} \setminus \mathcal{{\rm Ann}_R(M)}$‎. ‎Two distinct vertices $x‎, ‎y$ are connected whenever ${\rm Ann}_{\mathcal{M}}(xy)$ is an essential submodule‎ ‎of $\mathcal{M}$‎. ‎The present study provides a detailed analysis of planar zero-divisor and planar essential graphs‎, ‎especially when they possess a universal vertex‎. ‎It demonstrates that such graphs can be represented‎ ‎as join of some known graphs‎. ‎Additionally‎, ‎it examines that whether‎ ‎the zero-divisor and the essential graphs of $\mathbb{Z}_n$ are planar or not‎. ‎ zero-divisor graph Essential graph planar graph https://jart.guilan.ac.ir/article_9435_805e4e7165249a36185606713393cd2d.pdf