Bounding the parameter $β$ of a distance-regular graph with classical parameters

Abstract

Let $Γ$ be a distance-regular graph with classical parameters $(D, b, α, β)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(α,b)$ only depending on $α$ and $b$, such that if $β\geq C(α, b)r^2$, then either $Γ$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(α, b)$ only depending on $α$ and $b$, such that if $β\geq C_1(α, b)r$, then either $Γ$ is a Grassmann graph, or a bilinear forms graph.