The Structure of Generalized Cayley Graph When \(Cay(G,S) = P_2\) X \(P_2\) and \(P_2\) X \(C_3\)

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<=S. As a result, the vertices of the Cayley graph Cay (Ψ,S) are all members of Ψ, and two nearby vertices, x and y, are only adjacent if xy^−1 ∈S. The given generalized Cayley graph is defined as \(Cay_m(G,S)\) This is a graph whose vertex set is made up of every column matrix \(X_m\) It has two vertices and all of its components in Ψ. \(X_m\) and \(Y_m\) are adjacent ↔ \(X_m[(Y_m)^-1]^t ∈\) M(S), where \(Y_m^-1\) is a column matrix in which ∀ entry correlates to an associated element's inverse. Y-m and M(S) is a m×m matrix where every entry is in S ,[y^-1]^i is the opposite of y^-1 andM>=1 . In this study, we assign the structure of the new graph and highlight some of its fundamental aspects \(Cay_m(G,S)\)  when \(Cay(G,S)\) is the \(P_2\) X \(P_2\) and \(P_2\) X \(C_2\).