The co-prime power order graph of a finite group

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Mushatet H. H.; Talebi A. A.

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Journal Paper

Paper Information

Journal: Journal of Algebra and Related Topics Year:2025 | Volume:13 | Issue:1 Page(s): 45-52

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Information Journal Paper

Title

The co-prime power order graph of a finite group

Author(s)

Mushatet H. H. | Talebi A. A. | Issue Writer Certificate

Keywords

Cyclic group

Dihedral group

Generalized quaternion group

complete graph

planar graph

Co-prime order graph

Induced graph

Vertex-connectivity

Abstract

In this study‎, ‎we generalized the co-prime graph of a finite group called the co-prime power order graph of a finite group‎. ‎It is denoted by $\beta_G$‎, ‎and its vertex set is $G$‎, ‎such that two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{gcd}(|x|,|y|)=p^n$‎, ‎where $p$ is a prime number‎, ‎and $n \in \mathbb{Z}^{+} \cup\{0\}$‎. ‎We characterized complete graphs and planar graphs on the co-prime power order graphs‎, ‎and investigated some properties of graph $\beta_G$ for some groups such as Cyclic groups‎, ‎dihedral groups‎, ‎and the Generalized quaternion groups‎, ‎and obtained the Vertex-connectivity among them‎. ‎Finally‎, ‎we characterized some induced subgraphs of co-prime power order graph for some finite groups.

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