Power Cayley Graphs of Dihedral Groups with Certain Order

Abstract

Combination of the concepts of power graph and Cayley graph associated to groups has led to the introduction to two new variations of Cayley graph known as the union power Cayley graph and the intersection power Cayley graph. The set of vertices for both graphs consist of the elements of a finite group G. Consider any inverse-closed subset S of G, two vertices x and y are adjacent in the union power Cayley graph if 〖xy〗^(-1)∈S or if either one is an integral power of the other. Furthermore, x and y are adjacent in the intersection power Cayley graph if 〖xy〗^(-1)∈S and if either one is an integral power of the other. In this paper, the generalization of the union power Cayley graphs and the intersection power Cayley graphs of the dihedral groups with order 2n, for n≥3 and n=P^m;P is prime and m is a natural number, relative to a specific subset containing rotation elements in the groups is found. In addition, properties of these graphs including the clique numbers, vertex chromatic numbers, girths and diameters are computed. Finally, the characteristics of the graphs, whether they are connected, regular, complete, and planar are also determined.