The Structure of Generalized Cayley Graph When \ (Cay (G, S) = P₂\) X \ (P₂\) and \ (P₂\) X \ (C₃\)

This work aims to present the generalized Cayley graph and identify its structure in a few specific scenarios. Assume that Ψ is a finite-group and that S is a non-empty subset of Ψ. e ∉ S and S^-1=1. In this study, we assign the structure of the new graph and highlight some of its fundamental aspects \ (Cayₘ (G, S) \) when \ (Cay (G, S) \) is the \ (P₂\) X \ (P₂\) and \ (P₂\) X \ (C₂\).