This question has been asked before several times but all the answers I came across gave the example where the spectrum was considered without counting multiplicities of the eigenvalues.
So can we find non-isomorphic graphs with same spectrum (where spectrum includes the multiplicities of eigenvalues also)?
Obviously, if I am given the spectrum along with the eigenvectors then I can get back the matrix (adjacency/laplacian) and hence the graph. But if I skip the information about eigenvectors how much do I lose?
I am also asking this because in my little study so far, I have only seen how eigenvalues reveal information about graphs (like connectivity, bipartite, approximate max-cut, independent set/colouring, etc). I have not seen how eigenvectors can be helpful for understanding graphs except for the little fact that they can be used to give a "nice" embedding of the graph on a plane.
Best Answer
Can you give a reference for your claim in the first paragraph? When people talk about isospectral graphs this almost always means isospectral including multiplicities. The examples on the wikipedia article on isospectral graphs include multiplicities. Methods to generate isospectral graphs like Seidel switching include multiplicities. If anything you might get a new research question if you consider isospectrality without matching multiplicities.