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.

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