Abstract
In 2017, Qiao and Koolen showed that for any fixed integer $D\geqslant 3$, there are only finitely many such graphs with $\theta_{\min}\leqslant -\alpha k$, 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 $\theta_{\min} \leqslant -\frac{D-1}{D}k$ for $D =4,5$.
Highlights
The odd girth of a non-bipartite graph is the length of its shortest odd cycle
In [6], Qiao and Koolen classified non-bipartite distance-regular graphs with valency k, diameter D 3 and smallest eigenvalue θmin −k/2
Using Theorem 1, we will classify non-bipartite distance-regular graphs with valency k, diameter D and smallest eigenvalue θmin
Summary
Let Γ be a non-bipartite distance-regular graph with valency k, diameter D, odd girth g and smallest eigenvalue θmin. We will study non-bipartite distance-regular graphs with relatively small θmin compared with k. Let Γ be a non-bipartite distance-regular graph with valency k and odd girth g, having smallest eigenvalue θmin. The (2t + 1)-gon has valency k = 2, odd girth g = 2t + 1 and smallest eigenvalue θmin. In [6], Qiao and Koolen classified non-bipartite distance-regular graphs with valency k, diameter D 3 and smallest eigenvalue θmin −k/2. Using Theorem 1, we will classify non-bipartite distance-regular graphs with valency k, diameter D and smallest eigenvalue θmin. Let Γ be a non-bipartite distance-regular graph with valency k, diameter D and smallest eigenvalue θmin k.
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