Spectral properties of the Cartesian product of K3 □ Kg graphs.

This study explores the Laplacian spectral characteristics of the Cartesian product graph of K₃ □ K_g, with particular emphasis on the trace, energy, and the coefficients of its characteristic polynomial. We establish general expressions for the trace of powers of the Laplacian matrix and develop recursive formulas for the Laplacian coefficients using trace-based identities. The findings are validated for various values of g, and generalized to arbitrary cases. Additionally, we derive upper bounds and lower bounds for the Laplacian eigenvalues of the graph K₃ □ K_g