How close does this come to resolving the Graph Reconstruction Conjecture (https://en.wikipedia.org/wiki/Reconstruction_conjecture)? I notice that we get the square of the eigenvector element values rather than the values themselves, so I don’t think that in and of itself, this resolves the reconstruction conjecture, but it gives some pretty convincing additional evidence!
In order for the reconstruction conjecture to be false in the face of this, there would need to be two graphs G1, G2 such that they are non-isomorphic, but share the same spectrum, and where the eigenvectors differ only in the choice of sign for the elements (and of course, where it’s not simply that we have one vector that is the negative of the other).

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