Light tails and the Hermitian dual polar graphs

Juri\'{s}i\v{c} et al. conjectured that if a distance-regular graph $\Gamma$ with diameter $D$ at least three has a light tail, then one of the following holds: 1.$a_1 =0$; 2.$\Gamma$ is an antipodal cover of diameter three; 3.$\Gamma$ is tight; 4.$\Gamma$ is the halved $2D+1$-cube; 5.$\Gamma$ is a Hermitian dual polar graph $^2A_{2D-1}(r)$ where $r$ is a prime power. In this note, we will consider the case when the light tail corresponds to the eigenvalue $-\frac{k}{a_1 +1}$. Our main result is: Theorem Let $\Gamma$ be a non-bipartite distance-regular graph with valency $k \geq 3$ , diameter $D \geq 3$ and distinct eigenvalues $\theta_0 > \theta_1 > \cdots > \theta_D$. Suppose that $\Gamma$ is $2$-bounded with smallest eigenvalue $\theta_D = -\frac{k}{a_1 +1}$. If the minimal idempotent $E_D$, corresponding to eigenvalue $\theta_D$, is a light tail, then $\Gamma$ is the dual polar graph $^2A_{2D-1}(r)$, where $r$ is a prime power. As a consequence of this result we will also show: Theorem Let $\Gamma$ be a distance-regular graph with valency $k \geq 3$, diameter $D \geq 2$, $a_1 =1$ and $\theta_0 > \theta_1 > \cdots > \theta_D$. If $c_2 \geq5$ and $\theta_D = -k/2$, then $c_2 =5$ and $\Gamma$ is the dual polar graph $^2A_{2D-1}(2)$.