Trees with Matrix Weights: Laplacian Matrix and Characteristic-like Vertices

Swetha Ganesh, Sumit Mohanty

Published 2020-09-13Version 1

In \cite{Kirkland}, the authors obtained an alternative characterization of characteristic vertices for trees with positive weights on its edges via Perron values and Perron branches. It was also shown that the algebraic connectivity of a tree with positive edge weights can be expressed in terms of Perron values. In this article, we consider trees with matrix weights on its edges. In particular, we are interested in trees with the following classes of matrix edge weights: 1. positive definite matrix weights, 2. lower (or upper) triangular matrix weights with positive diagonal entries. For trees with above classes of matrix edge weights, we define Perron value and Perron branch. Further, building on the work of \cite{Kirkland}, we have shown the existence of vertices satisfying properties analogous to the properties of characteristic vertices of trees with positive edge weights in terms of Perron values and Perron branches, and we call such vertices as characteristic-like vertices. In this case, the eigenvalues of the Laplacian matrix are nonnegative, and we obtain a lower bound for the first non-zero eigenvalue of the Laplacian matrix in terms of Perron values. Furthermore, we also compute the Moore-Penrose inverse of the Laplacian matrix for trees with nonsingular matrix weights on its edges.