Some Properties of Coprime Graph of Dihedral Group D_2n When n is a Prime Power

Abstract

The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph.

On the Girth, Independence Number, and Wiener Index of Coprime Graph of Dihedral Group

“…In 2020, Alimon, et al [12] examined the Szeged and Wiener index of the coprime graph of dihedral group 𝐷 2𝑛 for several 𝑛. Then Gazir, et al [13] described some properties of a coprime graph of a dihedral group 𝐷 2𝑛 where 𝑛 is a prime power. In 2021, Syarifudin, et al [14] continued the research conducted by [13] observing the diameter, radius, and girth of the coprime graph of a dihedral group 𝐷 2𝑛 for 𝑛 in general.…”

“…Then Gazir, et al [13] described some properties of a coprime graph of a dihedral group 𝐷 2𝑛 where 𝑛 is a prime power. In 2021, Syarifudin, et al [14] continued the research conducted by [13] observing the diameter, radius, and girth of the coprime graph of a dihedral group 𝐷 2𝑛 for 𝑛 in general. However, there are still unexplained conditions of 𝑛 for girth.…”

“…Gazir, et al[13] andSyarifudin, et al [14] gave some theorems about the form of the coprime graph of dihedral group 𝐷 2𝑛 . These theorems are as follows.…”