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 $q\le 1$ and $D \ge 4$ that support a uniform structure with respect to x.