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(i) v(j)) of an edge v(i) v(j) in G(#) is defined as the i jth 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-w 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-w is signature similar to a non-negative matrix.