Let the Kneser graph K of a distance-regular graph $\Gamma$ be the graph on
the same vertex set as $\Gamma$, where two vertices are adjacent when they have
maximal distance in $\Gamma$. We study the situation where the Bose-Mesner
algebra of $\Gamma$ is not generated by the adjacency matrix of K. In particular,
we obtain strong results in the so-called `half antipodal' case.
Fiol, M.; Brouwer, A. Distance-regular graphs where the distance-d graph has fewer distinct eigenvalues. "Linear algebra and its applications", 2015, vol. 480, p. 115-126.