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?



Operation on Isospectral graphs
If $G\_i$ is cospectral to $H\_i$ ($i=1,2$), then the direct products, with adjacency matrices
$$
 A(G\_i)\otimes A(G\_2),\quad A(H\_i)\otimes A(H\_2)
$$
are cospectral, as are the Cartesian products with adjacency matrices
$$
 A(G\_1)\otimes I + I\otimes A(G\_2),\quad A(H\_1)\otimes I + I\otimes A(H\_2)
$$
If $G$ and $H$ are cospectral and regular, their complements are cospectral. 
It follows that, for regular graphs, the lexicographic product preserves the spectrum. 
Cvetkovic and his colleagues has a theory of what they refer to as NEPS, which generalizes these observations. 


If $G$ and $H$ are cospectral strongly regular graphs and $u\in V(G)$, $v\in V(H)$, then
$G\setminus u$ and $H\setminus v$ are cospectral (as are their complements).


There is no hope of any exhaustive answer to this question.


