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?