The n-th coprime probability and its graph for some dihedral groups

The coprime probability and graph have been studied for various groups by many researchers focusing on the generalization of the probability part. For the coprime graph, the types and properties of the graph have been investigated and the patterns that can be found within a group are analysed. The coprime probability of a group is defined as the probability that the order of a random pair of elements in the group are relatively prime or coprime. Meanwhile, the coprime graph can be explained as a graph whose vertices are elements of a group and two distinct vertices are adjacent if and only if the greatest common divisor of the order of the first vertex and order of the second vertex is equal to one. It was unfortunate that the exploration of probabilities and graphs of groups have not considered both the n-th coprime probability and its graph that ultimately became the target in this research. Hence, the newly defined terms are then used to find the generalizations of the n-th coprime probability and the n-th coprime graphs for some dihedral groups. The types and properties of the graphs are also discussed in this research.


Introduction
Graph theory is the study of graphs in which it is formed through vertices or nodes joined by edges. Also, graph theory has its own significance and can benefit many areas of research. For example, the Konisberg bridge problem is solved by using the concept of planar and Eulerian graph. Another example of the application of graph theory is from the study of Natarajan and Balaci [1] where graph theory is used to find the shortest routes in online network services by setting up the variances involved in the research as vertices and edges. Studies on graphs with different approaches have been conducted over the years and in this research, the focus is on the extension of the prime graph.
The prime graph of a group was first introduced by Williams [2] in 1981 as a graph having the prime numbers dividing the order of G as its vertices and two vertices x and y are joined by an edge if and only if G contains an element of order .
xy The prime graph has attracted the attention of many researchers that they began to explore and extend the study under different scopes of groups such as dihedral groups, p -group, and nonabelian metabelian groups.
Ma et al. [3] extended the study of prime graph to the coprime graph of a group and it is defined as a graph whose vertices are elements of G and two distinct vertices are adjacent if and only if the

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\ G e has also been introduced in [5] and it is defined as follows: two distinct vertices are adjacent whenever their orders are relatively non-coprime. In the paper, the properties of a graph of a group are obtained and at the same time, they also discussed on the relation between the non-coprime graph and the prime graph. Later on, another graph has been introduced named as the relative coprime graph. Abd Rhani et al. [6] defined the relative coprime graph of a group as a graph whose vertices are elements of G and two distinct vertices x and y are joined by an edge if and only if their orders are coprime and any of x or y is in H where . H G  Inspired by the research of the extension of prime graph, later, it was discovered that among those studies, no published work has been done related to probability. So, researchers took the opportunity to explore on the coprime probability for some finite groups. Hence, in 2018, Abd Rhani [7] introduced a probability named as the coprime probability of a group. In the paper, the author generalized the coprime probability for dihedral group, n D where n is odd and p -group where p is prime. The definition and the results are stated as follows: Definition 1 [7] Coprime Probability of a Group Let G be a finite group. For any , x y G ∈ the coprime probability of , G denoted as 1. x y = Proposition 1 [7] Let G be a finite p -group of order n p where 1. n ≥ Then   [8] continued the study of coprime probability by emphasizing on the nonabelian metabelian groups of order less than 24. They also published another paper as in [9] by introducing the relative coprime probability which is also an extension from the coprime probability. In the paper, their scope of group was also on the nonabelian metabelian groups of order less than 24. Definition 2 describes the relative coprime probability of a group. Definition 2 [9] Relative Coprime Probability of a Group Let G be a finite group and H be a subgroup of . G The relative coprime probability of G is defined as follows; The results from [9] showed that if the groups of nonabelian metabelian groups of order less than 24 have the same order then their relative coprime probability are the same but still this is only applicable to certain orders only.
Therefore, this paper concentrates on the extension of the coprime probability and the extension of the coprime graphs where the -th n coprime probability and -th n coprime graphs are formulated. Then, the generalization of the -th n coprime probability for dihedral group, p D where p is prime is determined. Besides, the types and properties of the -th n coprime graphs which include the diameter, chromatic number, domination number, and the independence number are also obtained.
Hence, this paper is structured as follows: the first part discusses on the introduction of this research while the second part states the basic concepts, definitions and proposition for both groups and graphs theory that are useful throughout this research. In the third and fourth sections, the main results and the conclusion of this paper for both the n th − coprime probability and the -th n coprime graphs for dihedral group, p D where p is prime are discussed.

Preliminaries
In this section, some basic concepts on groups and graphs are stated.

Definition 3 [10] Dihedral Groups of Degree n
For each n ∈  and 3, n ≥ n D denoted as the set of symmetries of a regular n -gon. Furthermore, the order of n D is 2 . n The dihedral groups, n D can be represented in a form of generators and relations given in the following representation: 2 1 , : , .

Definition 8 [13] The Domination Number
The dominating set ( ) X V ⊆ Γ is a set where for each v outside , X there exists x X ∈ such that v is adjacent to .
x The minimum size of X is called the dominating number and it is denoted by ( ) γ Γ .
Definition 9 [13] The Independence Number A non-empty set S of ( ) V Γ is called an independent set of Γ if there is no adjacent between two elements of S in .
Γ Thus, the independent number is the number of vertices in the maximum independent set and it is denoted as ( ) α Γ .
3. -th n Coprime Probability and -th n Coprime Graph A new probability named the -th n coprime probability, denoted as ( ) ncopr P G , is introduced in this paper which is also an extension from the coprime probability. The definition for this probability is stated and for further understanding of this definition, 3 D is used as an example to describe the -th n coprime probability.      Next, a new graph is also introduced named as the -th n coprime graph which is also an extension from the study of the prime graph. The definition is given below and 3 D is used an example to illustrate the definition.  Table 1 and 2, below are the graphs that can be formed. When 1, 2,3, 4,5, n = a complete tripartite graph is constructed whereas when 6, n = a star graph is formed.

Results and Discussions
This section is divided into two parts. The first part discussed the generalization of the -th n coprime probability which stated in Theorem 1. Then, the second part discussed the types of the -th n coprime graphs with the properties of their graph. Throughout this research, the study on both -th n coprime probability and its graphs are determined for dihedral group, p D where p is prime.

Theorem 1
Let G be a dihedral group, p D where p is prime. For any , ,  where p is prime. The types and some properties of the graph which include the diameter, the chromatic number, the domination number and the independence number are analyzed.

Proposition 3
Let G be a dihedral group, p D where p is prime. Then