Generalized cayley graphs and group structure: Insights from the direct products of P₂ and C₃

This work introduces a generalization of Cayley graphs, denoted Cay m (, S), where is a finite group and S is a non-empty subset of .In this construction, vertices are represented by m-dimensional column vectors with entries in , and adjacency is determined by a matrix-based condition involving the inverse elements and a matrix of elements from S. We focus on elucidating the structure and fundamental properties of Cay m (, S) when the classical Cayley graph Cay(, S) corresponds to the direct products P 2 P 2 and P 2 C 2 .Through rigorous analysis, we reveal distinct structural characteristics arising from these specific group structures and their associated generating sets.The generalized Cayley graph, denoted as Cay m (, S), is a graph where the vertex set consists of all column matrices X m , with each matrix having elements from the set . Two vertices X m and Y m are adjacent if and only if Y m -1 , the inverse of Y m , is a column matrix where each entry corresponds to the inverse of the associated element in .Our findings provide valuable insights into the interplay between algebraic properties of groups and the topological features of their generalized Cayley graph representations.This study contributes to a deeper understanding of generalized Cayley graphs and their potential applications in diverse fields such as network theory, coding theory, and cryptography.