The co-prime power order graph of a finite group

Document Type : Research Paper

Authors

1 Ministry of Education, Dhi Qar Education Directorate, Iraq

2 Department of Mathematics, University of Mazandaran, Babolsar, Iran

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.

Keywords

References

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Journal of Algebra and Related Topics
Volume 13, Issue 1
July 2025
Pages 45-52

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