The inverse of the distance matrix of a distance well-defined graph

A square matrix L is called a Laplacian-like matrix if Lj = 0 and j(T) L = 0. A square matrix D is left (or right) Laplacian expressible if there exist a number lambda not equal 0, a column vector beta satisfying beta(T)j = 1, and a square matrix L such that beta(T) D = lambda j(T), LD + I = beta j(T) and Lj = 0 (or D beta = lambda j, DL + I = j beta(T) and j(T) L = 0). We consider the generalized distance matrix D (see Definition 4.1) of a graph whose blocks correspond to left (or right) Laplacian expressible matrices. Then D is also left (or right) Laplacian expressible, and the inverse D-1, when it exists, can be expressed as the sum of a Laplacian-like matrix and a rank one matrix. (C) 2016 Elsevier Inc. All rights reserved.SCI(E)ARTICLE11-2951