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\noindent
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{\bf Chai Wah Wu}
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{\bf Multipartite Separability of Laplacian Matrices of Graphs}
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Recently, Braunstein et al. 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
Braunstein et al.  Finally, we show that the Laplacian matrix of any
product of graphs (strong, Cartesian, tensor, lexicographical, etc.)
is multipartite separable, extending analogous results for bipartite
and tripartite separability.



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