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Is there a class of graphs (besides the path graphs) for which we know that the Laplacian L = D - A (where D is the degree matrix and A is the adjacency matrix) has simple spectrum, i.e. all Laplacian eigenvalues have multiplicity 1?

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For a while, we were not sure you existed! – Baby Dragon Jul 12 '13 at 20:34
Roughly speaking, this is probably true for almost all graphs:… – Felix Goldberg Jul 22 '13 at 10:35

For a start, there's the complements of the paths. (If the Laplacian eigenvalues of a graph are all simple, then so are the eigenvalues of its complement.) Most regular graphs have only simple eigenvalues; in particular if my sage computations can be trusted then 6 of 21 cubic graphs on 10 vertices have only simple eigenvalues.

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Just to complement Chris's answer: Eigenspaces of $L$ are representation spaces of the automorphism group of the graph. So a graph with many symmetries is likely to have eigenvalues of high multiplicity. – Alain Valette Jul 12 '13 at 21:09

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