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We know that two graphs are isomorphic iff their adjacency matrices are similar via a permutation matrix. But if the adjacency matrices are similar, does it imply the graphs are isomorphic?

This question Any proof or counterexamples are welcomed, thank you!

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@QiaochuYuan Thanks!But I wonder how to prove that the same multiset of eigenvalues in adjacency matrices implies similarity. – Golbez Feb 24 '13 at 1:59
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If your graphs are undirected, this follows by the spectral theorem (which implies that adjacency matrices are diagonalizable). – Qiaochu Yuan Feb 24 '13 at 6:59
    
@QiaochuYuan Got it! You can make it an answer so that I can select as the best answer. – Golbez Feb 24 '13 at 9:30

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