The Hamming graph H (d, q) satisfies the following conditions: (i)For any 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.(ii)For any 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 a1 + 1. In this paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph. © 2007 Elsevier Ltd. All rights reserved.
Hiraki, A. (2008). A characterization of the Hamming graph by strongly closed subgraphs. European Journal of Combinatorics, 29(7), 1603–1616. https://doi.org/10.1016/j.ejc.2007.11.001