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

Abstract

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

10 pages. arXiv admin note: text overlap with arXiv:1711.05874