Let $G$ and $H$ be simple graphs.

I am interested in the Laplacian spectrum for various products of $G$ and $H$ namely the cartesian product, tensor product, lexicographical product and strong product as defined here https://en.wikipedia.org/wiki/Graph_product .

Given that $\lambda_1, \ldots, \lambda_n$ and $\mu_1, \ldots \mu_m$ are the eigenvalues of the Laplacians of $G$ and $H$ respectively, it is well known that the eigenvalues of the carteisan product of $G$ and $H$ are $$ \lambda_i + \mu_j \quad \hbox{for} \quad i = 1,\ldots,n \quad \hbox{and} \quad j = 1, \ldots,m.$$

I am interested in the relation between the eigenvalues of $G$ and $H$ with respect to the eigenvalues of the other mentioned products.

The same problem has already been considered for the spectrum of the adjacency matrix and solved under the general setting of the NEPS operation.

I suspect the same problem for the spectrum of the Laplacian eigenvalue to be slightly harder (as I think this would somehow have to characterize when is the lexicographical product of $G$ and $H$ connected) but I am not sure as I was not able to find any literature related to this matter.

Anyone happens to know the answer or could possibly provide some literature on this matter?