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Light tails and the Hermitian dual polar graphs

Juriśič et al. conjectured that if a distance-regular graph $Γ$ with diameter $D$ at least three has a light tail, then one of the following holds: 1.$a_1 =0$; 2.$Γ$ is an antipodal cover of diameter three; 3.$Γ$ is tight; 4.$Γ$ is the halved $2D+1$-cube; 5.$Γ$ 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 $Γ$ be a non-bipartite distance-regular graph with valency $k \geq 3$ , diameter $D \geq 3$ and distinct eigenvalues $θ_0 > θ_1 > \cdots > θ_D$. Suppose that $Γ$ is $2$-bounded with smallest eigenvalue $θ_D = -\frac{k}{a_1 +1}$. If the minimal idempotent $E_D$, corresponding to eigenvalue $θ_D$, is a light tail, then $Γ$ 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 $Γ$ be a distance-regular graph with valency $k \geq 3$, diameter $D \geq 2$, $a_1 =1$ and $θ_0 > θ_1 > \cdots > θ_D$. If $c_2 \geq5$ and $θ_D = -k/2$, then $c_2 =5$ and $Γ$ is the dual polar graph $^2A_{2D-1}(2)$.