An eigenvector interlacing property of graphs that arise from trees by Schur complementation of the Laplacian

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.