A characterization of the Hamming graph by strongly closed subgraphs

The Hamming graph H ( d , q ) satisfies the following conditions: (i) y pair ( u , v ) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance between u and v . In particular, any strongly closed subgraph is distance-regular. y pair ( x , y ) of vertices at distance d − 1 the subgraph induced by the neighbors of y at distance d from x is a clique of size a 1 + 1 . s paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph.