A characterization of the Hamming graph by strongly closed subgraphs

The Hamming graph H(d,q)H(d,q) satisfies the following conditions: (i)For any pair (u,v)(u,v) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance between uu and vv. In particular, any strongly closed subgraph is distance-regular.(ii)For any pair (x,y)(x,y) of vertices at distance d−1d−1 the subgraph induced by the neighbors of yy at distance dd from xx is a clique of size a1+1a1+1.In this paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph.