In addition to the good questions already given, let me add a reference to R. Hammack, W. Imrich, and S. Klav\v{z}ar, Handbook of Product Graphs, CRC 2011. I do not have the book at hand, but if I remember correctly the authors do suggest that each kind of product is essentially compatible with one, and just one, of the classical graph matrices (Laplacian, adjacency, normalised Laplacian are those they consider, if I'm not mistaken).
Delio Mugnolo
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