In this paper we find formulas for group inverses of Laplacians of weighted trees. We then develop a relationship between entries of the group inverse and various distance functions on trees. In particular, we show that the maximal and minimal entries on the diagonal of the group inverse correspond to certain pendant vertices of the tree and to a centroid of the tree, respectively. We also give a characterization for the group inverses of the Laplacian of an unweighted tree to be an M-matrix.
SIAM Journal on Matrix Analysis and Applications
Kirkland, S. J.; Neumann, M.; and Shader, Bryan L. (1997). "Distances in Weighted Trees and Group Inverse of Laplacian Matrices." SIAM Journal on Matrix Analysis and Applications 18.4, 827-841.