Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices

Recently, Laplacian matrices of graphs are studied as density matrices in quantum mechanics. We continue this study and give conditions for separability of generalized Laplacian matrices of weighted graphs with unit trace. In particular, we show that the Peres–Horodecki positive partial transpose condition is necessary and sufficient for separability in C2⊗CqC2⊗Cq. In addition, we present sufficient conditions for separability of generalized Laplacian matrices and diagonally dominant matrices.