title: On the Fiedler vectors of graphs that arise from trees by Schur complementation of the Laplacian.
abstract: The utility of Fiedler vectors in interrogating the structure of graphs has generated intense interest and motivated the pursuit of further theoretical results. This paper focuses on how the Fiedler vectors of one graph reveal structure in a second graph that is related to the first. Specifically, we consider a point of articulationrin the graphGwhose Laplacian matrix isLand derive a related graphG{r}whose Laplacian is the matrix obtained by taking the Schur complement with respect torinL. We show how Fiedler vectors ofG{r}relate to the structure ofGand we provide bounds for the algebraic connectivity ofG{r}in terms of the connected components atrinG. In the case whereGis a tree with points of articulationr∈R, we further consider the graphGRderived fromGby taking the Schur complement with respect toRinL. We show that Fiedler vectors ofGRvaluate the pendent vertices ofGin a manner consistent with the structure of the tree.
authors.affiliation: North Carolina State University.
authors.forename: Eric A
authors.initials: EA
authors.lastname: Stone
authors.forename: Alexander R
authors.initials: AR
authors.lastname: Griffing