**5. Discussion**

Laplacian spectra and their applications for different networks have been studied for many years, but the Laplacian spectra and their applications for networks based on graph operations are very rare. Laplacian spectra are used as a tool to analyze the structure of a network, and the importance of graph products cannot be denied in multilayer networking. Therefore, we have considered all cases of categorical product networks to compute Laplacian spectra and further to find their applications. The Laplacian spectra of a family of recursive trees and their applications in network coherence were discussed in [17]. Furthermore, we have computed the Laplacian spectra of the categorical product network and have established the expressions for the product of the nonzero Laplacian eigenvalues and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Using these expressions, we have

computed the Kirchhoff index, also called the "network criticality", for the categorical product network. The enumeration of spanning trees on generalized pseudofractal networks is discussed in [18]."
