Griffing Alexander R, Lynch Benjamin R, Stone Eric A
North Carolina State University.
Linear Algebra Appl. 2013 Feb 1;438(3):1078-1094. doi: 10.1016/j.laa.2012.10.005. Epub 2012 Nov 20.
The literature is replete with rich connections between the structure of a graph G = (V, E) and the spectral properties of its Laplacian matrix L. This paper establishes similar connections between the structure of G and the Laplacian L* of a second graph G*. Our interest lies in L* that can be obtained from L by Schur complementation, in which case we say that G* is partially-supplied with respect to G. In particular, we specialize to where G is a tree with points of articulation r ∈ R and consider the partially-supplied graph G* derived from G by taking the Schur complement with respect to R in L. Our results characterize how the eigenvectors of the Laplacian of G* relate to each other and to the structure of the tree.
文献中充满了关于图(G = (V, E))的结构与其拉普拉斯矩阵(L)的谱性质之间的丰富联系。本文建立了(G)的结构与第二个图(G^)的拉普拉斯矩阵(L^)之间的类似联系。我们感兴趣的是通过舒尔补从(L)得到的(L^)，在这种情况下，我们说(G^)相对于(G)是部分提供的。特别地，我们专门研究(G)是具有关节点(r ∈ R)的树的情况，并考虑通过对(L)中关于(R)取舒尔补从(G)导出的部分提供的图(G^)。我们的结果刻画了(G^)的拉普拉斯特征向量之间的相互关系以及与树结构的关系。
