Distance-regular graphs with classical parameters that support a uniform structure: case $q \le 1$

Abstract

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid \partial(x,y) = \partial(x,z)\}$, where $\partial$ denotes the path-length distance in $\Gamma$. Observe that the graph $\Gamma_f=(X,\mathcal{R}_f)$ is bipartite. We say that $\Gamma$ supports a uniform structure with respect to $x$ whenever $\Gamma_f$ has a uniform structure with respect to $x$. Assume that $\Gamma$ is a distance-regular graph with classical parameters $(D,q,\alpha,\beta)$ with $q \le 1$. Recall that $q$ is an integer, which is not equal to $0$ or $-1$. The purpose of this paper is to study when $\Gamma$ supports a uniform structure with respect to $x$. The main result of the paper is a complete classification of graphs with classical parameters with $q\leq 1$ and $D \ge 4$ that support a uniform structure with respect to $x$.