It is a well known fact that there are non-isomorphic connected graphs whose adjacency matrix have the same spectrum.
This has been discussed, for example, in this older post.
However, in the examples given, the graph spectrum always some has repeated eigenvalues. Are there examples where this is not the case, but rather each eigenvalue is unique?
On a different but related problem: Is it true that if two graphs have only single eigenvalues (multiplicity 1) in their spectrum, then it is easy to determine if they are isomorphic or not?