  The relationships between eigenvalues and eigenvectors of a product graph and
those of its factor graphs have been known for the standard products, while
characterization of Laplacian eigenvalues and eigenvectors of the Kronecker
product of graphs using the Laplacian spectra and eigenvectors of the factors
turned out to be quite challenging and has remained an open problem to date.
Several approaches for the estimation of Laplacian spectrum of the Kronecker
product of graphs have been proposed in recent years. However, it turns out
that not all the methods are practical to apply in network science models,
particularly in the context of multilayer networks. Here we develop a practical
and computationally efficient method to estimate Laplacian spectra of this
graph product from spectral properties of their factor graphs which is more
stable than the alternatives proposed in the literature. We emphasize that a
median of the percentage errors of our estimated Laplacian spectrum almost
coincides with the $x$-axis, unlike the alternatives which have sudden jumps at
the beginning followed by a gradual decrease for the percentage errors. The
percentage errors confined (confidence of the estimations) up to $\pm$10% for
all considered approximations, depending on a graph density. Moreover, we
theoretically prove that the percentage errors becomes smaller when the network
grows or the edge density level increases. Additionally, some novel theoretical
results considering the exact formulas and lower bounds related to the certain
correlation coefficients corresponding to the estimated eigenvectors are
presented.
