A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs |
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Authors: | Akira Hiraki |
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Affiliation: | 1. Division of Mathematical Sciences, Osaka Kyoiku University, Osaka, 582-8582, Japan
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Abstract: | 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. |
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