The Clique Number and The Chromatics Number Of The Coprime Graph for The Generalized Quarternion Group

ABSTRACT


A. INTRODUCTION
Graph theory is a useful tool to describe a real-world problem as a mathematics problem such as the in schedulling problem, in chemical graph topological indices, Jahandideh et al. (2015) or in atom bond connectivity index Hua et al. (2019), so the solution can be found easily. In recent years, a graph can give a representation of abstract mathematical systems such as groups or rings (Zavarnitsine, 2006). With graphs, we can give meaning to groups or rings, such as the visualization of groups or rings and we can define the distance between the elements of groups or rings. We have many graph representations for a group, such as the coprime graph Alimon et al. (2020), the non-coprime graph Mansoori et al. (2016), the intersection graph Akbari et al. (2015), and the power graph (Aşkin & Büyükköse, 2021).
There have been several studies regarding coprime graphs from finite groups such as coprime graphs and non-coprime graphs from the generalized quaternion group Nurhabibah et al. (2021), the dihedral group , the integer modulo group Series & Science (2021) and representation of non-coprime graphs from an integer modulo (Misuki et al., 2021) and non-coprime graph of the generalized quaternion group (Nurhabibah et al., 2022). Other popular studies in graph representation are the intersection graph for the dihedral group Ramdani et al. (2022), the prime graph Satyanarayana (2010) and the power graph for the dihedral group Asmarani et al. (2021) and integer modulo group (Syechah et al., 2022). Based on the study of the coprime graph on generalized quaternion group and the search for clique number and chromatic number (Husni et al., 2022) it can be analyzed further about the properties of the graph. In this study, the authors analyze the clique number and chromatic number of the coprime graph of the generalized quaternion group ( 4 ).

B. METHODS
In this study, the authors searched various literature related to generalized quaternion groups, coprime graphs, as well as clique numbers, and chromatic numbers. Then analyze several examples so that a certain pattern is obtained which is then expressed as a conjecture. The conjecture is then proved to obtain its truth value. If the conjecture proves to be true, then it is stated as a theorem and if not, then the writer will construct a new conjecture until the correct conjecture is obtained.

C. RESULT AND DISCUSSION
In this research, the writer determines the clique number and chromatic number of the coprime graph of the generalized quaternion group ( 4 ). The generalized quaternion group ( 4 ) is one of the finite groups with the following definition.

Definition 1
The generalized quaternion group ( 4 ) is a 4 order group composed of two elements ( , ) or can be written 4 = { , | 2 = , 2 = , −1 = −1 } where is the identity element with ≥ 2. This group can be represented in several types of graphs, one of which is a coprime graph. Suppose is a finite group, the coprime graph of is denoted by 4 are vertices with elements of and two different vertices and are adjacent if and only if ( | |. | |) = 1 (Ma et al., 2014). Some results regarding the coprime graph of the generalized quaternion group ( 4 ) are given in the following 4 theorems . Theorem 1 Suppose 4 is a generalized quaternion group, if = 2 then the coprime graph of 4 is completely bipartite.
Theorem 2 Suppose 4 is a generalized quaternion group, if = with odd primes, then the coprime graph of 4 is tripartite .
Theorem 4 Suppose 4 is a generalized quaternion group, if = 1 1 2 2 3 3 … … with ≠ 2, then the coprime graph of 4 is + 2 -. The first discussion in this study is the clique number of the coprime graph of the generalized quaternion group ( 4 ) (Nurhabibah et al., 2022).

Clique Number
The clique number is one of the properties of a graph which is defined based on the complete subgraphs in the graph. A formal definition of the clique number is given in Definition 2 below.

Definition 2
The clique number ( ) of graph is the maximum order among complete subgraphs in .
The definition of a complete graph is contained in Definition 3 as follows Definition 3 A complete graph is a simple graph in which every vertex has edges to all other vertices (Nurhabibah et al., 2022). A complete graph with vertices is denoted by .
The following is an example of the clique number of the coprime graph of the generalized quaternion group ( 4 ). The graph in figure 1 has several complete subgraphs and the maximum order of these complete subgraphs is 3, so based on Definition 2, ( 4.3 ) = 3. With an analysis similar to Example 1, the author derives several theorems about the clique numbers of the coprime graph of the group generalized quaternion.
Proof: Let 4 be a coprime graph. Take = 2 with ∈ ℕ. We will show ( 4 ) = 2. This means that it will be shown that there is 2 which is a complete subgraph 4 and there is no complete subgraph with > 2. The author obtains a complete subgraph 2 from 4 which is a graph with ( 2 ) = { , 1 } ∈ 4 with 1 ∈ 4 \{ }. Suppose that there is a complete subgraph of 4 with > 2. This means that 4 must be a − graph with > 2. This contradicts Theorem 1 which states that 4 is a complete bipartite graph. So 2 is a complete subgraph of 4 of maximal order. It is proved that ( 4 ) = 2. ∎ Theorem 6 below is the clique number of the coprime graph of the group generalized quaternion ( 4 ) with = where is an odd prime number.
Proof: Let 4 be a coprime graph. Take = with odd primes. It will show ( 4 ) = 3. This means that it will be shown that there is 3 which is a complete subgraph 4 and there is no complete subgraph with > 3. The author obtains a complete subgraph 3 from 4 which is a graph with ( 3 ) = { , 1 , 2 } ⸦ 4 with 1 ∈ 1 where 1 is the set of all vertices in 4 of order and 2 ∈ 2 where 2 is the set of all vertices in 4 with even order. Suppose that there is a complete subgraph of 4 with > 3. This means that 4 must be a − graph with > 3. This contradicts Theorem 2 which states 4 is a tripartite graph.
So 3 is a complete subgraph of 4 with maximum order, it is proved that ( 4 ) = 3.
Proof: Let 4 be a coprime graph. Take = 1 1 2 2 3 3 … … with ≠ 2 . Will show ( 4 ) = + 2. This means that it will be shown that there exists +2 which is a complete subgraph of 4 and there is no complete subgraph with > + 2. The writer obtains a complete subgraph +2 from 4 is a graph with ( +2 ) = { , , 1 , 2 , … . , } ∈ 4 with ∈ where R is the set of all vertices in 4 with even order and ∈ . Suppose that there are complete subgraphs of 4 with > + 2. This means that 4 must be a − graph with > + 2. This contradicts Theorem 4 which states 4 + 2 − graph.
So +2 is a complete subgraph of 4 with maximal order, it is shown that ( 4 ) = + 2. ∎ The second discussion in this study is the chromatic number of the coprime graph of the generalized quaternion group ( 4 ).

Chromatic Number
Besides the clique number, the chromatic number is also one of the properties of a graph which is defined based on the coloring of the graph vertices. A formal definition of a chromatic number is given in Definition 4 below.

Definition 4
The chromatic number of a graph is the minimum number of colors needed to color all the vertices of such that every two neighboring vertices get a different color . The chromatic number of a graph , denoted by ( ).
The following is an example of the chromatic number of the coprime graph of the generalized quaternion group ( 4 ).
Example 2: For example, for = 3, the coprime graph of the generalized quaternion group ( 4 ) is as shown in Figure 2. Based on the image, the minimum color needed to color the graph in figure 2 is 3, so ( 12 ) = 3. With more or less the same steps, the authors derive several theorems about the chromatic number of the coprime graph of the generalized quaternion group ( 4 ).
Proof: Let 4 be a coprime graph. Take = 2 with ∈ . It will be shown that ( 4 ) = 2. This means that it will be shown that the minimum number of colors needed to color the vertices of 4 so that every two neighboring vertices with different colors is 2. According to Theorem 1, 4 is a complete bipartite graph, namely a star graph. The partition consists of 2 subsets, namely 1 = { }, 2 = { 4 } \ { } . Notice that node e is next to every node in 2 . So that every vertex in 2 cannot have the same color as { }. Also, note that none of the vertices in 2 are adjacent to each other so that the color of each vertex in 2 is the same. So, the minimum number of colors required is 2. It is proved that ( 4 ) = 2. ∎ Theorem 10 Let 4 be a coprime graph of 4 . If = with p odd prime numbers. Then ( 4 ) = 3.
Proof: Let 4 be a coprime graph. Take = with p odd primes. It will show ( 4 ) = 3. This means that it will be shown that the minimum number of colors needed to color the vertices of 4 so that every two neighboring vertices with different colors is 3. According to Theorem 2, 4 is a tripartite graph. The partition consists of 3 subsets, namely { , 1 , 2 } with Theorem 11 below is the chromatic number of the coprime graph of the group generalized quaternion ( 4 ) with = 1 1 2 2 3 3 … … .

D. CONCLUSION AND SUGGESTIONS
The clique number and chromatic number of the coprime graph of the generalized quaternion group ( 4 ) are 2,3, + 1, and + 2.
There are many open problem that can be brought up after this result, such as the topological indices of the coprime graph of the generalized quaternion group.