Bounding the parameter ${\hat{I}}^2$ of a distance-regular graph with classical parameters

Abstract

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