The M-Matrix Group Generalized Inverse Problem for Weighted Trees

We characterize all weighted trees whose Laplacian has a group inverse which is an M-matrix. Actually, only a very narrow set of weighted trees yields such Laplacians. Our investigation involves analyzing circumstances under which a certain Z-matrix, derived from the tree and whose order is one less than the number of vertices in the tree, is an M-matrix. Our work here is motivated by a recent paper of Styan and Subak-Sharpe [ Linear Algebra Appl., 250 (1997), pp. 349--370] and by an earlier question of Deutsch and Neumann [J. Math. Anal. Appl., 102 (1984), pp. 1--29].