We know that the Laplacian matrix of a simple graph can be diagonalized as $L = U\Lambda U^T$ where $U$ shows eigenvectors and $\Lambda$ contains the eigenvalues. Can the Laplacian of two different graphs (unweighted and undirected) have the same eigenvectors and their difference is in the eigenvalues? If so, is there any relation between these two graphs?
I think eigenvectors of the Laplacian of all the circulant graphs are the same, but I can't prove it or find any proof for it. Is it right?