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We study entanglement properties of mixed density matrices obtained fromcombinatorial Laplacians. This is done by introducing the notion of the densitymatrix of a graph. We characterize the graphs with pure density matrices andshow that the density matrix of a graph can be always written as a uniformmixture of pure density matrices of graphs. We consider the von Neumann entropyof these matrices and we characterize the graphs for which the minimum andmaximum values are attained. We then discuss the problem of separability bypointing out that separability of density matrices of graphs does not alwaysdepend on the labelling of the vertices. We consider graphs with a tensorproduct structure and simple cases for which combinatorial properties arelinked to the entanglement of the state. We calculate the concurrence of allgraph on four vertices representing entangled states. It turns out that forsome of these graphs the value of the concurrence is exactly fractional.Comment: 20 pages, 11 figure

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States