A distance-regular graph with bipartite geodetically closed subgraphs

Let Γ be a distance-regular graph of diameter d , and t be an integer with 2 ≤ t ≤ d − 1 such that at−1 = 0. For any pair (u, v) of vertices, let Π (u, v) be the subgraph induced by the vertices lying on shortest paths between u and v. We prove that if Π (u, v) is a bipartite geodetically closed subgraph for some pair (u, v) of vertices at distance t , then Π (x, y) is a bipartite geodetically closed subgraph for any pair (x, y) of vertices at distance less than or equal to t . In particular, Π (x, y) is either a path, an ordinary polygon, a hyper cube or a projective incidence graph. © 2003 Elsevier Science Ltd. All rights reserved.