中文摘要：

为了得出一类密度矩阵的可分判据研究了特殊图,利用图理论、拉普拉斯矩阵性质、部分转置正判据、图上顶点与其部分转置图上对应顶点之间的度数关系,分别给出了完全纠缠（perfect entangled,PE）匹配图在C^pC^q与C^3C^4量子系统中的可分判据.证明了在C^pC^q量子系统中,若n=pq个顶点上的PE-匹配图的部分转置不是PE-匹配的,则该图的密度矩阵是纠缠的,否则其部分转置是非负（positive partialtranspose,PPT）的;并给出了C^3C^4系统中n=3×4个顶点上的PE-匹配图的密度矩阵可分的充要条件是该图的部分转置也是PE-匹配图.

英文摘要：

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 C^3C^4 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 C^3C^4 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.