M-Matrix Inverse problem for distance-regular graphs

Abstract

We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is a M–matrix;that is, it has non–positive off–diagonal elements or, equivalently when the Moore-Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M–property. We prove that only distance–regular graphs with diameter up to three can have the M–property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover we exhaustively analyze the strongly regular graphs having the M-property and we give some families of distance regular graphs with diameter three that satisfy the M-property.