Generalized Coprime Graph of Subgroups of a Finite Group

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DOI: https://doi.org/10.28924/2291-8639-24-2026-306
Published: Oct 5, 2026
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Bushra N. Abdulgaphur, Sanhan M.S. Khasraw, Rashad R. Haji, Generalized Coprime Graph of Subgroups of a Finite Group, Int. J. Anal. Appl., 24 (2026), 306.

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Bushra N. Abdulgaphur, Sanhan M.S. Khasraw, Rashad R. Haji

Abstract

Let \(\mathcal{G}\) be a finite group. The generalized coprime graph of \(\mathcal{G}, \Gamma_{g c}(\mathcal{G})\), is a graph whose vertices are all the proper non-trivial subgroups of \(\mathcal{G}\) and two distinct vertices \(H\) and \(K\) are adjacent if \(|H| \nmid|K|\) and \(|K| \nmid|H|\). The paper investigates the connectedness, diameter and girth of \(\Gamma_{g c}(\mathcal{G})\). Also, we explore the interplay between the algebraic properties of \(\mathcal{G}\) and the graph theoretic properties of \(\Gamma_{g c}(\mathcal{G})\). Furthermore, the energy of \(\Gamma_{g c}\left(D_{2 n}\right)\), where \(n \in\left\{2 p, p^{k}, p q\right\}\), and \(\Gamma_{g c}\left(\mathbb{Z}_{n}\right)\), where \(n \in\left\{p q r, p^{k} q, p^{2} q^{2}\right\}\), is computed.

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