Uniqueness of distance-regular graphs with specified classical parameters

Determine whether, for sufficiently large diameter \(D\), the Grassmann graphs \(J_2(2D+1,D)\) and the twisted Grassmann graphs \(\widetilde{J}_2(2D+1,D)\) are the only distance-regular graphs with classical parameters \((D,2,2,2^{D+2}-2)\).

Background

The paper applies its Hoffman-graph structural theory to local graphs of distance-regular graphs with classical parameters. In the special case $(D,2,2,2^{D+2}-2)$, the results provide structural information about every local graph.

The authors note that two infinite families are currently known with these parameters: the Grassmann graphs $J_2(2D+1,D)$ and the twisted Grassmann graphs $\widetilde{J}_2(2D+1,D)$. It remains unresolved whether these known families exhaust all distance-regular graphs with the specified parameters when the diameter is sufficiently large.