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

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