We　investigate　separability　of　Laplacian　matrices　of　graphs　when　seen　as　density　matrices.　This　is　a　family　of　quantum　states　with　many　combinatorial　properties.　We　firstly　show　that　the　well-known　matrix　realignment　criterion　can　be　used　to　test　separability　of　this　type　of　quantum　states.　The　criterion　can　be　interpreted　as　novel　graph-theoretic　idea.　Then,　we　prove　that　the　density　matrix　of　the　tensor　product　of　N　graphs　is　N-separable.　However,　the　converse　is　not　necessarily　true.　Additionally,　we　derive　a　sufficient　condition　for　N-partite　entanglement　in　star　graphs　and　propose　a　necessary　and　sufficient　condition　for　separability　of　nearest　point　graphs.