Generalized Coprime Graph of Subgroups of a Finite Group

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

doi:10.28924/2291-8639-24-2026-306

Research Article

Generalized Coprime Graph of Subgroups of a Finite Group

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

* Corresponding author

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.

How to Cite

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

BibTeX

@article{Abdulgaphur2026,
  author  = {Abdulgaphur, Bushra N. and Khasraw, Sanhan M.S. and Haji, Rashad R.},
  title   = {Generalized Coprime Graph of Subgroups of a Finite Group},
  journal = {Int. J. Anal. Appl.},
  volume  = {24},
  year    = {2026},
  pages   = {306},
  doi     = {10.28924/2291-8639-24-2026-306}
}

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