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.