Prime order and composite order Cayley graphs of generalised quaternion group and quasi-dihedral group

Abstract

A Cayley graph is a structure consisting of vertices and edges that describes the information of a group and its generators where two vertices are connected by a directed edge in certain conditions. This research focuses on the prime order and composite order Cayley graphs on the generalised quaternion group and the quasidihedral group, where the subsets are the set of prime order and composite order element of each group. The properties of elements of both groups are investigated, and then the structures of prime order and composite order Cayley graphs of generalised quaternion group and quasi-dihedral group are obtained. Besides, the properties of the graph such as chromatic number, independence number, clique number, diameter, girth, and graph planarity are found. From this research, it is shown that the prime order and composite order Cayley graphs of both generalised quaternion group and quasi-dihedral group consist of unions of isomorphic components of equal vertices and a regular connected graph, respectively.