Smallest non-isomorphic graphs with the same characteristic polynomials of their laplacian matrices

Question

For isomorphic graphs, the characteristic polynomials of their laplacian matrices coincide, but the converse is not true. The characteristic polynomial of the laplacian matrix does not uniquely identify the graph.

What are the smallest non-isomorphic graphs with coinciding characteristic polynomials of their laplacian matrices ?

Answer 1

There is a pair of Lapcian cospectral graphs on six vertices. See http://www.math.ucsd.edu/~fkenter/cospectral_talk.pdf