Connectivity Indices of Coprime Graph of Generalized Quarternion Group

Abstract

Generalized quarternion group (Q_(4n)) is a group of order $4n$ that is generated by two elements x and y with the properties x^{2n}=y^4=e and xy=yx^{-1}. The coprime graph of Q_{4n}, denoted by Omega_{Q_{4n}}, is a graph with the vertices are elements of Q_{4n} and the edges are generated by two elements that have coprime order. The first result of this paper presented that Omega_{Q_{4n}} is a tripartite graph for n is odd and Omega_{Q_{4n}} is a star graph for n is even. The second one presented the connectivity indices of Omega_{Q_{4n}}. 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.Generalized quaternion group (Q4n) is a group of order 4n that is generated by two elements x and y with the properties x 2n = y 4 = 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.