Improving diameter bounds for distance-regular graphs

In this paper we study the sequence ( c i ) 0 ≤ i ≤ d for a distance-regular graph. In particular we show that if d ≥ 2 j and c j > 1 then c 2 j − 1 > c j holds. Using this we give improvements on diameter bounds by A. Hiraki, J.H. Koolen [An improvement of the Ivanov bound, Ann. Comb. 2 (2) (1998) 131–135], and L. Pyber [A bound for the diameter of distance-regular graphs, Combinatorica 19 (4) (1999) 549–553], respectively, by applying this inequality.