On the Moore-Penrose inverse of distance–regular graphs

We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M–matrix and then we say that the 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, in terms of their intersection array, of those distance–regular graphs that satisfy the M–property. It is remarkable that either a primitive strongly regular graph or its complement has the M–property.