Linear estimation in models based on a graph Original Research Article

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 formula for the Moore–Penrose inverse of the incidence matrix of a tree. An inequality involving the Moore–Penrose inverse of the Laplacian matrix of a graph 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.