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

Abstract: In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, 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 $\theta_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.