Perron values and classes of trees

Abstract

The bottleneck matrix $M$ of a rooted tree $T$ is a combinatorial object encoding the spatial distribution of the vertices with respect to the root. The spectral radius of $M$, known as the Perron value of the rooted tree, is closely related to the theory of the algebraic connectivity. In this paper, we investigate the Perron values of various classes of rooted trees by making use of combinatorial and linear-algebraic techniques. This results in multiple bounds on the Perron values of these classes, which can be straightforwardly applied to provide information on the algebraic connectivity.