Non-bipartite distance-regular graphs with a small smallest eigenvalue

In 2017, Qiao and Koolen showed that for any fixed integer D >= 3, there are only finitely many non-bipartite distance-regular graphs with theta(min) <= -alpha k, where 0 < alpha < 1 is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small theta(min) compared with k. In particular, we will show that if theta(min) is relatively close to -k then the odd girth g must be large. Also we will classify the non-bipartite distance-regular graphs with theta(min) <= -D-1/Dk for D = 4, 5.