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

Abstract 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.