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.
Azlesh, S. , Payrovi, S. and Soheilnia, F. (2026). Planarity of the essential graph for modules. Journal of Algebra and Related Topics, (), -. doi: 10.22124/jart.2025.30608.1799
MLA
Azlesh, S. , , Payrovi, S. , and Soheilnia, F. . "Planarity of the essential graph for modules", Journal of Algebra and Related Topics, , , 2026, -. doi: 10.22124/jart.2025.30608.1799
HARVARD
Azlesh, S., Payrovi, S., Soheilnia, F. (2026). 'Planarity of the essential graph for modules', Journal of Algebra and Related Topics, (), pp. -. doi: 10.22124/jart.2025.30608.1799
CHICAGO
S. Azlesh , S. Payrovi and F. Soheilnia, "Planarity of the essential graph for modules," Journal of Algebra and Related Topics, (2026): -, doi: 10.22124/jart.2025.30608.1799
VANCOUVER
Azlesh, S., Payrovi, S., Soheilnia, F. Planarity of the essential graph for modules. Journal of Algebra and Related Topics, 2026; (): -. doi: 10.22124/jart.2025.30608.1799
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