Light tails and the Hermitian dual polar graphs

Abstract

Juriśič et al. (Eur J Comb 31:1539–1552, 2010) conjectured (see also [van Dam et al. in Distance-regular graphs, Problem 13]) 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; and (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 first main result is:

Theorem Let $\Gamma$ be a non-bipartite distance-regular graph with valency $k \ge 3$, diameter $D \ge 3$ 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 $r=a_1+1$ is a prime power.

As a consequence of this result we will show our second main result:

Theorem Let $\Gamma$ be a distance-regular graph with valency $k \ge 3$, diameter $D \ge 2$, $a_1 =1$ and $\theta _0 > \theta _1 > \cdots > \theta _D$. If $c_2 \ge 4$ and $\theta _D = -k/2$, then $c_2 =5$ and $\Gamma$ is the dual polar graph $^2A_{2D-1}(2)$.