  It is known that there is an alternative characterization of characteristic
vertices for trees with positive weights on their edges via Perron values and
Perron branches. Moreover, the algebraic connectivity of a tree with positive
edge weights can be expressed in terms of Perron value.
  In this article, we consider trees with matrix weights on their edges. More
precisely, 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 the above classes
of matrix edge weights, we define Perron values and Perron branches. Further,
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
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 value. Furthermore, we
also compute the Moore-Penrose inverse of the Laplacian matrix of a tree with
nonsingular matrix weights on its edges.
