Akira Hiraki

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. © 2011 Springer.

Let Gamma be a Delsarte set graph with an intersection number c(2) (i.e., a distance-regular graph with a set C of Delsarte cliques such that each edge lies in a positive constant number n(C) of Delsarte cliques in C). We showed in Bang et al. (J Combin 28:501-506, 2007) that if psi(1) > 1 then c(2) >= 2 psi(1) where psi(1) := vertical bar Gamma(1)(x)boolean AND C vertical bar for x is an element of V(Gamma) and C aDelsarte clique satisfying d(x, C) = 1. In this paper, we classify Gamma with the case c(2) = 2 psi(1) > 2. As a consequence of this result, we show that if c(2) <= 5 and psi(1) > 1 then Gamma is either a Johnson graph or a folded Johnson graph (J) over bar (4s, 2s) with s >= 3.

In this paper we study a distance-regular graph Gamma of diameter d >= 4 such that for any given pair of vertices at distance d - 1 there exists a strongly closed subgraph of diameter d - 1 containing them. We prove several inequalities for intersection numbers of Gamma. We show that if the equalities hold in some of these inequalities, then Gamma is either the Odd graph, the doubled Odd graph, the doubled Grassmann graph, the Hamming graph or the dual polar graph. (C) 2008 Elsevier Ltd. All rights reserved.

Let Gamma 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: (SC)(m) : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. (BB)(m) : Let(x, y, z) be a triple of vertices with partial derivative(Gamma)(x, y) = 1 and partial derivative(Gamma)(x, z) = partial derivative(Gamma)(y, z) = m. Then B(x, z) = B(y, z). (CA)(m) : Let (x, y, z) be a triple of vertices with partial derivative(Gamma)(x, y) = 2, partial derivative(Gamma)(x, z) = partial derivative(Gamma)(y, z) = m and vertical bar C(z, x) boolean AND C(z, y)vertical bar >= 2. Then C(x, z) boolean OR A(x, z) = C(y, z) boolean OR A(y, z). 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 a(1) = 0 < a(2) 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, alpha, beta) such that c(2) > 1 and a(1) = 0 < a(2) satisfies the condition (SC)(i) for i = 1, ..., d - 1. In particular, either (b, alpha, beta) = (-2, -3, -1 - (-2)(d)) or (b, alpha, beta) = (-3, -2, -1+(-3)(d)/2) holds.

Let Gamma 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: (SC)(m) : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. (BB)(m) : Let (x, y, z) be a triple of vertices with partial derivative(Gamma) (x, y) = 1 and partial derivative(Gamma) (x, z) = partial derivative(Gamma) (y, z) = m. Then B(x, z) = B(y, z). (CA)(m) : Let (x, y, z) be a triple of vertices with partial derivative(Gamma) (x, y) = 2, partial derivative(Gamma) (x, z) = partial derivative(Gamma) (y, z) = m and |C(z, x) boolean AND C(z, y)| >= 2. Then C(x, z) boolean OR A(x, z) = C(y, z) boolean OR A(y, z). 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 Gamma.