Group inverses of weighted trees

Let (G,w) be an undirected weighted graph. The group inverse of (G,w) is the weighted graph with the adjacency matrix A#, where A is the adjacency matrix of (G,w). We study the group inverse of singular weighted trees. It is shown that if (T,w) is a singular weighted tree, then T# is again a tree, if and only if T is a star tree, which in turn, holds if and only if T# is graph isomorphic to T. A new class T of weighted trees, is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in T, is signature similar to a non-negative matrix.