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Let Γ be a distance-regular graph of diameter d ≥ 3 with c 2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions: Suppose that the condition ( SC) m holds. Then it has been known that the condition ( BB) i holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition ( CA) i holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions ( BB) i and ( CA) i hold for all i with 1 ≤ i ≤ m, then the condition ( SC) m holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph in Γ. [ABSTRACT FROM AUTHOR]

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ i = |{ x | ∂( u, x) = ∂( v, x) = 1 and ∂( w, x) = i − 1}| ( i = 2, 3,..., d) depends only on i whenever ∂( u, v) = 2 and ∂( u, w) = ∂( v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph. [ABSTRACT FROM AUTHOR]