A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

摘要

In this paper we study a distance-regular graph Γ of diameter d ≥ 3 which satisfies the following two conditions: (a) For any integer i with 1 ≤ i ≤ d − 1 and for any pair of vertices at distance i in Γ there exists a strongly closed subgraph of diameter i containing them; (b) There exists a strongly closed subgraph Δ which is completely regular in Γ. It is known that if Δ has diameter 1, then Γ is a regular near polygon. We prove that if a strongly closed subgraph Δ of diameter j with 2 ≤ j ≤ d − 1 is completely regular of covering radius d − j in Γ, then Γ is either a Hamming graph or a dual polar graph.