# Generalized order divisor graphs of finite group

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Tech- nology Thanyaburi (RMUTT), 12110, Pathum Thani, Thailand

2 Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, 12120, Pathum Thani, Thailand

10.22108/ijgt.2022.133091.1787

Abstract

Let $G$ be a finite group and $k$ a fixed positive integer. We define the generalized order divisor graph of $G$ to be a graph whose vertex set is the group $G$ and in which two vertices $a$ and $b$ are adjacent if and only if the orders $o(a^k)$ and $o(b^k)$ are different and either $o(a^k)$ divides $o(b^k)$ or $o(b^k)$ divides $o(a^k)$. This generalizes the order divisor graphs of finite groups. Some properties of our graph are introduced, and we investigate the structure of the generalized order divisor graphs of finite cyclic groups.

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