Non-identity order divisor graphs of groups

Sattanathan, M.; Kottarathil, J.; Chakrabarty, I.

doi:10.22124/jart.2025.28373.1716

Non-identity order divisor graphs of groups

Articles in Press

Document Type : Research Paper

Authors

¹ Department of Mathematics, Sri Paramakalyani College, Alwarkurichi,Tamil Nadu, India

² Department of Mathematics, St. Joseph's College, Moolamattom, Kerala, India

³ Department of Mathematics, CHRIST(Deemed to be University), Bengaluru, India

10.22124/jart.2025.28373.1716

Abstract

Let $G$ be a group with identity $e$. In this paper, we define and study the non-identity order divisor graph of $G$, where the vertex set is $G-\{e\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if either $O(x)|O(y)$ or $O(y)|O(x)$. We denote the order divisor graph of group $G$ by $\o(G)$. We study some basic properties of $\o(G)$ such as connectedness, completeness, bipartiteness and Eulerian property. The lower bound as well as the number of edges of $\o(G)$ are also calculated for some group $G$ and some characterizations for fundamental properties of $\o(G)$ have been obtained. Finally, we explore the relation between the order prime graph and the non-identity order divisor graph of some group $G$.

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