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 jTL=0. A square matrix D is left (or right) Laplacian expressible if there exist a number λ≠0, a column vector β satisfying βTj=1, and a square matrix L such that βTD=λjT, LD+I=βjT and Lj=0 (or Dβ=λj, DL+I=jβT and jTL=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.