The intersection power Cayley graph of cyclic groups of order 𝒑𝒒

One of the modern approaches to exploring group properties is by observing the relationship among its elements using graph theory. In this manuscript, a new graph namely the intersection power graph is introduced and constructed for cyclic groups of order 𝑝𝑞, ℤ_𝑝𝑞, where 𝑝 is odd prime number and 𝑞 = 2 with respect to subsets of a certain size of ℤ_𝑝𝑞. In the structure of Cayley graphs of cyclic groups, the diameters of the intersection power graph structure decrease linearly with the number of vertices. The intersection power Cayley graph of a finite group 𝐺 related to the subset 𝑆 of 𝐺, is a graph in which the elements of 𝐺 are the vertices and 𝑥 and 𝑦 are adjacent if 𝑥 = 𝑔𝑦 or 𝑦 = 𝑔𝑥 for some 𝑔 ∈ 𝑆 and if either one is an integral power of the other. Through this manuscript, the general presentations for the intersection power Cayley graph on cyclic groups of order 𝑝𝑞, where 𝑝 is odd prime number and 𝑞 = 2 with respect to subsets of a certain size of ℤ_𝑝𝑞 are obtained. Besides, some properties of the graph, which include connectivity, regularity, completeness, and planarity are determined in this paper. Moreover, the invariants of the graphs such as clique number, vertex chromatic number, girth and diameter are also computed. Finally, analogous results for Euler’s function are achieved, particularly for specifying the number of elements whose relationship is prime with a cyclic group of composite order.