Separability of PE-matching Graph

The separable criterion of a class of density matrix is presented by studying a special graph.Using graph theory,the property of Laplacian matrix,the positive partial transpose criterion and the relationship of degree between the vertices of graph and the corresponding vertices of partial transpose of the graph,the separable criterion of PE-matching graph in C^p⊗C^q and C³⊗C⁴ is given respectively.In C^p⊗C^q quantumsystems,It is proven that if the partial transpose of a PE-matching graph on n = pq verticesis not a PE-matching,thedensity matrix of this graph is entanglement,otherwise it is PPT(positive partial transpose).It is also presented that in C³⊗C⁴ systems if the density matrix of PEmatching graph on n = 3 x 4 vertices is separable,the necessary and sufficient condition is that the partial transpose of this graph is also a PE-matching graph.