The The Cayley Graph of Semi-Direct Product of finite Groups: Interrelationships and Construction

In this paper, we study Cayley graph of the semi-direct product of two finite groups where is an odd prime numbe. Specifically, we endeavor to establish a comprehensive understanding of the Cayley graph by investigating the interrelationships among the constituent elements of the group. Leveraging this acquired knowledge, we systematically construct the Cayley graph, there by contributing to the scholarly discourse on this subject matter.