On Terwilliger Graphs in Which the Neighborhood of Each Vertex is Isomorphic to the Hoffman–Singleton Graph

The Hoffman–Singleton graph is the only strongly regular graph with parameters $(50,7,0,1)$. A well-known hypothesis states that a distance-regular graph in which the neighborhood of each vertex is isomorphic to the Hoffman–Singleton graph has intersection array $\{50,42,1;1,2,50\}$ or $\{50,42,9;1,2,42\}$. In the present paper, we prove this hypothesis under the condition that a distance-regular graph is a Terwilliger graph and the graph diameter is at most $5$.