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arXiv · 1901.01157

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

Zhi Qiao·Yifan Jing·Jack Koolen
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Abstract

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $θ_{\min}\leq -αk$, where $0<α<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $θ_{\min}$ compared with $k$. In particular, we will show that if $θ_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $θ_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.

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Zhi Qiao, Yifan Jing, Jack Koolen. 2019-01-03. Non-bipartite distance-regular graphs with a small smallest eigenvalue. https://arxiv.org/abs/1901.01157

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