Light tails and the Hermitian dual polar graphs

Abstract

Juriśič 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 ${}^2{A}_{2D-1}(r)$ 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 $k\ge 3$ , 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 ${}^2{A}_{2D-1}(r)$, 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 $k\ge 3$, diameter $D\ge 2$, $a_1=1$ and ${\theta }_0>{\theta }_1>\cdots >{\theta }_D$. If $c_2\ge 5$ and ${\theta }_D=-k/2$, then $c_2=5$ and $\Gamma$ is the dual polar graph ${}^2{A}_{2D-1}(2)$.

References in corpus (1)

• Distance-regular graphs with valency k having smallest eigenvalue at most -k/2