  Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of
graphs as density matrices in quantum mechanics and studied the relationships
between quantum physical properties and graph theoretical properties of the
underlying graphs. We provide further results on the multipartite separability
of Laplacian matrices of graphs. In particular, we identify complete bipartite
graphs whose normalized Laplacian matrix is multipartite entangled under any
vertex labeling. Furthermore, we give conditions on the vertex degrees such
that there is a vertex labeling under which the normalized Laplacian matrix is
entangled. These results address an open question raised in [1]. Finally, we
extend some of the results in [1,2] to the multipartite case and show that the
Laplacian matrix of any product of graphs (strong, Cartesian, tensor,
categorical, etc.) is multipartite separable.
