Diameter bounds for geometric distance-regular graphs

A non-complete distance-regular graph is called geometric if there exists a set C of Delsarte cliques such that each edge lies in exactly one clique in C. Let Gamma be a geometric distance regular graph with diameter D >= 3 and smallest eigenvalue theta(D). In this paper we show that if Gamma contains an induced subgraph K-2,K-1,K-1, then D <= theta(D). Moreover, if -theta(D) - 1 <= D <= theta(D) then D = -theta(D) and Gamma is aJohnson graph. We also show that for (s, b) is not an element of{(11,11), (21, 21)}, there are no distance -regular graphs with intersection array {4s, 3(s-1), s+1-b; 1, 6, 4b} where s, b are integers satisfying s >= 3 and 2 <= b <= s. As an application of these results, we classify geometric distance-regular graphs with D >= 3, theta(D) -4 and containing an induced subgraph K-2,K-1,K-1. (C) 2017 Elsevier B.V. All rights reserved.