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I have looked everywhere for a straightforward answer, but there seems to be no explanation for what you can do with the eigenvalues of a graph.

What properties about the graph like connectivity, modeling it as a network, etc. can you tell from the eigenvalues?

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    Some answers at https://math.stackexchange.com/questions/308952/motivation-for-spectral-graph-theory/312555#312555 – Chris Godsil May 11 '17 at 21:03

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