Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics

Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum systems. In this note we look at the graphs whose normalized Laplacian matrices are separable or entangled. In particular, we show that the number of such graphs is related to the number of 0–1 matrices that are line sum symmetric and to the number of graphs with at least one vertex of degree 1.