Distance-regular graphs where the distance-d graph has fewer distinct eigenvalues

Let the Kneser graph K of a distance-regular graph Γ be the graph on the same vertex set as Γ, where two vertices are adjacent when they have maximal distance in Γ. We study the situation where the Bose–Mesner algebra of Γ is not generated by the adjacency matrix of K. In particular, we obtain strong results in the so-called ‘half antipodal’ case.