But what about the reverse problem? Given $n$ eigenvalues, can one (efficiently) find a graph that has this spectra?
I suspect that in general this is hard to do (and might be equivalent to GI) but what if you relax some of the conditions a bit? What if you make conditions that there is no multiplicity of eigenvalues? What about allowing graphs that have "close" spectra by some distance metric?
Any references or ideas would be welcome.
As Suresh points out, if you allow undirected weighted graphs with self loops, this problem becomes pretty trivial. I was hoping to get answers on the set of undirected, unweighted simple graphs but I would be happy with simple unweighted directed graphs as well.