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

TL;DR

This paper investigates non-bipartite distance-regular graphs with small smallest eigenvalues, establishing relationships between eigenvalues, odd girth, and classifying certain graphs based on eigenvalue bounds.

Contribution

It provides new classifications of non-bipartite distance-regular graphs with specific eigenvalue bounds and explores the relationship between eigenvalues and graph girth.

Findings

Large odd girth when smallest eigenvalue is close to -k

Classification of graphs with eigenvalues ≤ (D-1)/D for D=4,5

Finiteness results for graphs with eigenvalues below a threshold

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 $D =4,5$.