Linear estimation in models based on a graph

Two natural linear models associated with a graph are considered. The Gauss -Markov theorem is used in one of the models to derive a combinatorial form ula for the Moore-Penrose inverse of the incidence matrix of a tree. An ine quality involving the Moore-Penrose inverse of the Laplacian matrix of a gr aph and its distance matrix is obtained. The case of equality is discussed. Again the main tool used in the proof is the theory of linear estimation. (C) 1999 Elsevier Science Inc. All rights reserved.