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.