THE UNION POWER CAYLEY GRAPHS AND INTERSECTION POWER CAYLEY GRAPHS OF CYCLIC GROUPS AND DIHEDRAL GROUPS WITH THEIR CLASSIFICATIONS AND INVARIANTS

Abstract

Various graphs associated to groups have been investigated and defined over the years, including the power graphs and Cayley graphs, due to their importance in algebra and many other fields. The power graph of a finite group 𝐺 is defined as a simplified form of an undirected graph whose vertices are elements of 𝐺, in which two distinct vertices are adjacent if one of them can be written as an integral power of the other. Meanwhile, the Cayley graph of 𝐺 with respect to the inverse-closed subset 𝑆 of 𝐺 is a graph whose vertices are the elements of 𝐺, and two vertices 𝑥 and 𝑦 are adjacent if 𝑥 = 𝑠𝑦 or 𝑦 = 𝑠𝑥 for some 𝑠 ∈ 𝑆. In this research, two new types of Cayley graphs are introduced by combining the properties of Cayley graphs and power graphs, namely the union power Cayley graph and the intersection power Cayley graph of a finite group. The union power Cayley graph is defined as a graph that has the elements of 𝐺 as its vertices, and two vertices 𝑥 and 𝑦 are adjacent if 𝑥𝑦−1 ∈ 𝑆 or if one of them can be written as an integral power of the other. Meanwhile, the intersection power Cayley graph is defined as a graph whose vertices are the elements of 𝐺, and two vertices 𝑥 and 𝑦 are adjacent if 𝑥𝑦−1 ∈ 𝑆 and if one of them can be written as an integral power of the other. In addition to introducing these two new graphs, this research also aims to classify these graphs in terms of their connectivity, completeness, regularity, and planarity and to determine the invariants of these graphs, including the clique, chromatic number, diameter, and girth. The theoretical results provided in this research are significant in the development of algebraic graph theory since the invariants of finite groups can be identified from the structure of these graphs. The results of this research are obtained by finding the general presentations for the union power Cayley graph and the intersection power Cayley graph of cyclic groups 𝐶𝑛, and .dihedral groups