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 $A^{\mathrm{\#}}$, 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 $T^{\mathrm{\#}}$ is again a tree, if and only if $T$ is a star tree, which in turn, holds if and only if $T^{\mathrm{\#}}$ is graph isomorphic to $T$. A new class $\mathbb{T}$ 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 $\mathbb{T}$, is signature similar to a non-negative matrix.