CONNECTIVITY INDICES OF COPRIME GRAPH OF GENERALIZED QUATERNION GROUP

Generalized quaternion group (Q4n) is a group of order 4n that is generated by two elements x and y with the properties x2n = y4 = e and xy = yx−1. The coprime graph of Q4n, denoted by ΩQ4n , is a graph with the vertices are elements of Q4n and the edges are formed by two elements that have coprime order. The first result of this paper presents that ΩQ4n is a tripartite graph for n is an odd prime and ΩQ4n is a star graph for n is a power of 2. The second one presents the connectivity indices of ΩQ4n . Connectivity indices of a graph is a research area in mathematics that popularly applied in chemistry. There are six indices that are presented in this paper, those are first Zagreb index, second Zagreb index, Wiener index, hyper-Wiener index, Harary index, and Szeged index.


INTRODUCTION
Graph theory has been widely applied in many fields. One of them is in chemistry, which is related to connectivity indices. Connectivity indices are molecular descriptor which is computed based on the molecular graph of chemical compound. The molecular graph can be assumed as a graph. There are some kinds of connectivity indices that are interesting to be discussed. Such indices are hyper-Wiener, Harary, the first Zagreb, the second Zagreb, and Szeged. Some of these indices can be used to analyze the chemical properties of paraffines [9].
There are many kinds of research relating to graph and group theory since the properties of a group can be easily seen when a graph represents that group. There are some previous results that have been discussed related to connectivity indices of a graph, especially in mathematics. Some of the results are determining connectivity indices of non-commuting graph of dihedral group (D 2n ) [1] and generalized quaternion group (Q 4n ) [8]. Other than the non-commuting graph, another kind of graph represents a group, namely a coprime graph. Recently, not many studies have learned connectivity indices of the graph associated with groups. Moreover, the quaternion group has similar properties to the dihedral group. Therefore we are interested in studying the connectivity indices of the coprime graph of Q 4n .

PRELIMINARIES
In this section we present some definitions that are needed in this study.
Definition 2.1. [11] Let n be a natural number. The generalized quaternion group, denoted by Q 4n , is defined as Hence the order of Q 4n is 4n.
Let G be a finite group and g ∈ G. The order of g, denoted by |g|, is the smallest natural number n such that g n = e, where e is an identity element of G.

Definition 2.4. [2]
Let k be a natural number, a graph Ω is a k−partite graph if its vertex set, V (Ω) can be partitioned into k subsets V 1 , V 2 , ..., V k such that every edge of Ω joins vertices in two different partite sets. A 2−partite graph is called bipartite and 3−partite graph is called tripartite.

Definition 2.5. [2]
A complete k−partite graph Ω is a k−partite graph that two vertices are adjacent in Ω if and only if the vertices belong to different partite sets. If |V i | = n i for 1 i k, then Ω is denoted by K n1,n2,...,n k . For complete bipartite graph K 1,n is also called a star graph, denoted by S n . Definition 2.7. [3] Let Ω be a simple connected graph. The first Zagreb index of Ω, denoted by M 1 (Ω), is defined as where deg(v) is degree of vertex v, i.e. the number of edges that incident to v.
Definition 2.8. [3] Let Ω be a simple connected graph. The second Zagreb index of Ω, denoted by M 2 (Ω), is defined as Definition 2.9. [4] Let Ω be a simple connected graph. The Wiener index of Ω, denoted by W (Ω), is defined as where d(u, v) is the distance between vertex u and v, i.e. the number of edges in shortest path connecting u and v.
Definition 2.10. [10] Let Ω be a simple connected graph. The hyper-Wiener index of Ω, denoted by W W (Ω), is defined as where d(u, v) is the distance between vertex u and v.
Definition 2.11. [10] Let Ω be a simple connected graph. The Harary index of Ω, denoted by H(Ω), is defined as where d(u, v) is the distance between vertex u and v.

RESULTS AND DISCUSSIONS
This section consists of two subsections. The first subsection discuss about the coprime graph of Q 4n and the second one discuss about its connectivity indices.
3.1. Coprime Graph of Generalized Quaternion Group. In this subsection, we determine the coprime graph of Q 4n . Since the adjacency of the vertices depends on the order of elements of Q 4n , then firstly we determine the order of elements of Q 4n on the following lemma.
Lemma 3.1. Let Q 4n be a generalized quaternion group. Then the order of its elements are showed as follows , then there are two cases. Case 1. The order of elements x i y j for 0 < i < 2n and j = 0. According to Definition 2.1 we have Let m = 2, by induction we will show that The proof is similar to case 2b, hence it is omitted.
The proof is similar to case 2b, hence it is omitted.
The next result is the shape of the coprime graph of generalized quaternion group that presented in the following theorem.
Theorem 3.2. Let Q 4n be a generalized quaternion group and Ω Q4n be the coprime graph of generalized quaternion group. Then i. Ω Q4n is a tripartite graph for n is an odd prime ii. Ω Q4n is a star graph for n is a power of 2.
Proof. Since |e| = 1 and |x i y j | = 1 for i, j = 0, then clearly that vertex e is adjacent to any other vertices of Ω Q4n .
Then the partition of V (Ω Q4n ) is {{e}, S, T }. According to Lemma 3.1, each vertex x 2k ∈ S has the same order, i.e. n, which means any two vertices in S are not adjacent. For vertices in T , we divide into three groups, those are vertex x n , vertex set x 2k+1 for 1 k n − 1, and vertex set x i y for 0 i 2n − 1. Therefore from Lemma 3.1 we have gcd(|x n |, |x 2k+1 |) = 1, gcd(|x n |, |x i y|) = 1, and gcd(|x 2k+1 |, |x i y|) = 1 which means any vertices in T is not adjacent to each other. Since n is an odd prime, then gcd(|x 2k |, |x n |) = 1. Hence S and T cannot be in the same partition. Thus Ω Q4n is a tripartite graph. For n is an odd prime, we redefine the vertex set and edge set of Ω Q4n to make easier in determining its connectivity indices. Let Based on the enumerate above, we can illustrate Ω Q4n as follows: Graph ΩQ 4n for n is an odd prime.

Connectivity
Indices. The connectivity indices of Ω Q4n are determined on the following results.
Proof. Let n be an odd prime. Firstly we determine the degree of each vertex of Ω Q4n based on Therefore Let n be a power of 2.
Let n be a power of 2.
Since Ω Q4n is a star graph S 4n−1 , then the distance between vertex e and any other vertices on Ω Q4n is one and the distance is two for any two vertices in V (Ω Q4n ) − {e}. Therefore Theorem 3.7. Let Ω Q4n be the coprime graph of generalized quaternion group. The hyper-Wiener index of Ω Q4n is W W (Ω Q4n ) = 20n 2 − 12n + 4, for n is an odd prime 24n 2 − 14n + 2, for n is a power of 2.
Proof. Let n be an odd prime. Firstly we determine the sum of square of the distance between any two vertices of Ω Q4n as follows Referring Theorem 3.6 we have Let n be a power of 2. According to Theorem 3.6 and its proof, we have Proof. Since the Harary index is the summation of inverse of distances between any two vertices of Ω Q4n , then we can determine it based on the proof of Theorem 3.6. Therefore, for n is an odd prime we have For n is a power of 2 we have Theorem 3.9.
Proof. Let n be an odd prime. From Figure 3.1 we can determine the vertices of Ω Q4n which are closer to one of two vertices that are adjacent as follows: (i) Edge a = ex n .

Conclusion
In this study, we have found the shape of coprime graph of generalized quaternion group and determined its six connectivity indices.