Group Inverses of Weighted Trees

Let (G, w) be a weighted graph with the adjacency matrix A. The group inverse of (G, w), denoted by (G^#, w^#) is the weighted graph with the weight w^#(v<inf>i</inf>v<inf>j</inf>) of an edge v<inf>i</inf>v<inf>j</inf> in G^# is defined as the ijth entry of A^# , the group inverse of A. We study the group inverse of singular weighted trees. It is shown that if (T, w) is a singular weighted tree, then (T^#, w^#) is again a weighted tree if and only if (T, w) is a star tree, which in turn holds if and only if (T^#, w^#) is graph isomorphic to (T, w). A new class T<inf>w</inf> 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<inf>w</inf> is signature similar to a non-negative matrix.