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 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. 

