Cospectral graphs are graphs having same eigenvalues. Constructions of cospectral graph is an interesting question in graph theory. Now a days we use graph theory in different brunches of Sciences and Technology. I am looking for some applications of the idea of cospectral graph. Please provide some examples with references.
I'll add some thoughts partially in response to Igor's answer, in that while I agree that cospectral graphs are intrinsically interesting, I think there is a bit more to it than that.
Many authors (including mathematicians, chemists, physicists) over many years have tried to develop complete graph invariants that can be computed in polynomial time, but a good fraction of these turn out to be spectral invariants in disguise (with respect to some matrix or other). So then cospectral graphs are useful counterexamples.
(These papers often proceed along the lines: here's an invariant obtained by counting walks or something similar in some complicated way. I tried it on lots of small graphs and it works, so I conjecture it always works. A few non-isomorphic strongly regular graphs with the same parameters will often usefully stress test these algorithms.)
More generally, there is the question of how much, and what, information about a graph can be deduced from its spectrum. Cospectral graphs tell you what can't - for example, number of edges can be deduced, but degree sequence can't.
Well, this paper: ISOSPECTRAL GRAPHS AND MOLECULES W. C. HERNDON* and M. L. ELLZEY, JR. (in Tetrahedron) describes some applications of isospectrality in chemistry; this seems to be the main application area (google application of isospectral graphs).
That said, I think mostly isospectral graphs are interesting for their own sake.