  Normalized Laplacian matrices of graphs have recently been studied in the
context of quantum mechanics as density matrices of quantum systems. Of
particular interest is the relationship between quantum physical properties of
the density matrix and the graph theoretical properties of the underlying
graph. One important aspect of density matrices is their entanglement
properties, which are responsible for many nonintuitive physical phenomena. The
entanglement property of normalized Laplacian matrices is in general not
invariant under graph isomorphism. In recent papers, graphs were identified
whose entanglement and separability properties are invariant under isomorphism.
The purpose of this note is to characterize the set of graphs whose
separability is invariant under graph isomorphism. In particular, we show that
this set consists of $K_{2,2}$, $\overline{K_{2,2}}$ and all complete graphs.
