TITLE: Operation on Isospectral graphs
QUESTION [5 upvotes]: Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that are obtained by applying a binary operation to $G$ and $H$?
For example, to take one special case, is $G‎\otimes‎G$ (Kronecker product) isospectral?
Which binary operations between $G$ and $H$ preserve the isospectrality?

REPLY [3 votes]: Chris seems to have forgotten that he and I published a generalization of the NEPS in
C. D. Godsil and B. D. McKay, Constructing cospectral graphs, Aequationes Mathematicae, 25 (1983) 257-268. This contains some general methods of using the tensor product, as well as some other techniques.