Some characterizatsion of coprime graph of dihedral group D _2n

Abstract

The Coprime graph of group G denoted Γ G is a graph with vertices is an element of G, and two distinct vertices are adjacent when its order relative prime. In 2020, Gazir et al. give some characterizations of Γ D 2n for n a prime power. The method that uses in this paper is deductive proof by taking some example of a coprime graph of D 2n , then generalized the characterization of example. This paper gives some characteristics of the coprime graph of a dihedral group for more general cases. One of the result, Γ D 2n is a multipartite graph with girth 3, radius 1, and diameter 2.

The Harmonic Index and the Gutman Index of Coprime Graph of Integer Group Modulo With Order of Prime Power

“…Definition 2 [1] The coprime graph of 𝐺 group, denoted by Γ 𝐺 is a graph whose vertices are elements of 𝐺 and two distinct vertices 𝑢 and 𝑣 are adjacent if and only if (|𝑥|, |𝑦|) = 1.…”

“…In recent years, a graph is used to visualize the algebraic structure such as groups or rings. Some of the graphs visualizing groups are the coprime graph of dihedral groups [1] and quaternion groups [2] on the dihedral group [3]. This coprime graph is introduced by Ma [4], and later the dual of the coprime graph, called non-coprime, that introduced by Mansoori [5], which also studied integer modulo [6] and dihedral group [7].…”

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

“…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.…”

“…In this paper, we will provide some properties for the form, girth, independence number, and the Wiener index of the coprime graph of dihedral group 𝐷 2𝑛 that have not been given in [12], [14] and [15].…”

“…Note that, the girth of graph Γ, denoted by 𝑔𝑖𝑟𝑡ℎ(Γ), is defined as the length of the shortest cycle in Γ.The following theorem given in[14] is about the girth of the coprime graph 𝐷 2𝑛 . 𝐷 2𝑛 ) = 3.…”