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 Γ be a geometric distance-regular graph with diameter D ≥ 3 and smallest eigenvalue θ D . In this paper we show that if Γ contains an induced subgraph K 2 , 1 , 1 , then D ≤ − θ D . Moreover, if − θ D − 1 ≤ D ≤ − θ D then D = − θ D and Γ is a Johnson graph. We also show that for ( s , b ) ⁄ ∈ { ( 11 , 11 ) , ( 21 , 21 ) } , there are no distance-regular graphs with intersection array { 4 s , 3 ( s − 1 ) , s + 1 − b ; 1 , 6 , 4 b } 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 , θ D ≥ − 4 and containing an induced subgraph K 2 , 1 , 1 . [ABSTRACT FROM AUTHOR]