Distance-Regular Graphs with a Relatively Small Eigenvalue Multiplicity

Abstract

Godsil showed that if $\Gamma$ is a distance-regular graph with diameter $D \geq 3$ and valency $k \geq 3$, and $\theta$ is an eigenvalue of $\Gamma$ with multiplicity $m \geq 2$, then $k \leq\frac{(m+2)(m-1)}{2}$.In this paper we will give a refined statement of this result. We show that if $\Gamma$ is a distance-regular graph with diameter $D \geq 3$, valency $k \geq 2$ and an eigenvalue $\theta$ with multiplicity $m\geq 2$, such that $k$ is close to $\frac{(m+2)(m-1)}{2}$, then $\theta$ must be a tail. We also characterize the distance-regular graphs with diameter $D \geq 3$, valency $k \geq 3$ and an eigenvalue $\theta$ with multiplicity $m \geq 2$ satisfying $k= \frac{(m+2)(m-1)}{2}$.

Distance-regular graphs

“…Koolen, Kim, and Park [411] refined the above valency bound of Godsil. By using the theory developed by Jurišić et al [387], they were able to show that for k ≥ 3 and m ≥ 3, the only possible distance-regular graphs with diameter at least 3 and k = (m+2)(m−1) 2 are Taylor graphs with intersection array {(2α + 1) 2 (2α 2 + 2α − 1), 2α 3 (2α + 3), 1; 1, 2α 3 (2α + 3), (2α + 1) 2 (2α 2 + 2α − 1)}, with m = 4α 2 + 4α − 1, where α is an integer not equal to 0 and −1 or α = −1± √ 5 2 (m = 3).…”