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Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that obtain by binary operation between $G$ and $H$? For example,in special case, is $G‎\otimes‎G$(Kronecker product) isospectral? which binary operations between $G$ and $H$, preserve the isospectrality?

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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.

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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.

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There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down. (Erdos) –  Gordon Royle Dec 18 '11 at 23:31
    
Actually, I believe it was Stan Ulam, not Erdos, who said that. –  Igor Rivin Dec 18 '11 at 23:44
    
See en.wikiquote.org/wiki/Stanislaw_Ulam –  Igor Rivin Dec 18 '11 at 23:50
    
Perhaps this will remind Chris: homepages.cwi.nl/~mueller/haemers.pdf –  Igor Rivin Dec 18 '11 at 23:53
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Since the eigenvalues of the tensor products of two matrices are products of the eigenvalues, the answer to your question would seem to be "yes". More interestingly it would appear that $G\otimes H$ and $H\otimes G$ would be isospectral for any graphs $G, H.$ (where by $G\otimes H$ I mean the graph whose adjacency matrix is the tensor product of those of $G, H.$)

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But $G\otimes H$ and $H\otimes G$ are isomorphic, and so their cospectrality is not a surprise. –  Chris Godsil Dec 18 '11 at 22:14
    
@Chris: umm, indeed... –  Igor Rivin Dec 18 '11 at 22:24
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