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Let Γ be a distance-regular graph of diameter d ≥ 3 with c2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions: In [12] we have shown that the condition ( SC) m holds if and only if both of the conditions ( BB) i and ( CA) i hold for i = 1,..., m. In this paper we show that if a1 = 0 < a2 and the condition ( BB) i holds for i = 1,..., m, then the condition ( CA) i holds for i = 1,..., m. In particular, the condition ( SC) m holds. Applying this result we prove that a distance-regular graph with classical parameters ( d, b, α, β) such that c2 > 1 and a1 = 0 < a2 satisfies the condition ( SC) i for i = 1,..., d − 1. In particular, either ( b, α, β) = (− 2, −3, −1 − (−2) d) or $$(b,\alpha,\beta)= \left(-3,-2,-\frac{1+(-3)^{d}}{2} \right)$$ holds. [ABSTRACT FROM AUTHOR]

Let Γ be a distance-regular graph of diameter d ≥ 3 with c2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions: In [12] we have shown that the condition ( SC) m holds if and only if both of the conditions ( BB) i and ( CA) i hold for i = 1,..., m. In this paper we show that if a1 = 0 < a2 and the condition ( BB) i holds for i = 1,..., m, then the condition ( CA) i holds for i = 1,..., m. In particular, the condition ( SC) m holds. Applying this result we prove that a distance-regular graph with classical parameters ( d, b, α, β) such that c2 > 1 and a1 = 0 < a2 satisfies the condition ( SC) i for i = 1,..., d − 1. In particular, either ( b, α, β) = (− 2, −3, −1 − (−2) d) or $$(b,\alpha,\beta)= \left(-3,-2,-\frac{1+(-3)^{d}}{2} \right)$$ holds. [ABSTRACT FROM AUTHOR]

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]

In this paper we consider the number t of columns (1,k-2,1) in the intersection array of a distance-regular graph. We prove that t is at most one if k≥58. [ABSTRACT FROM AUTHOR]

Let Γ be a distance-regular graph of diameter d and valency k > 2. Suppose there exists an integer s with d ≤ 2s such that c i = b d-i for all 1 ≤ i ≤ s. Then Γ is an antipodal double cover. [ABSTRACT FROM AUTHOR]

We show that the number of columns (1, k − 2, 1) in the intersection array of distance-regular graphs is at most 20. [ABSTRACT FROM AUTHOR]

Let Г be a distance-regular graph of diameter d and i-th valency k. We show that if k = k for 2 + j ≥ d and 2 < j, then Г is a polygon ( k = 2) or an antipodal 2-cover ( k = 1). We also give a short proof of Terwilliger's inequality for bipartite distance-regular graphs and a refinement of Ivanov's argument on diameter bound. [ABSTRACT FROM AUTHOR]