Distance-regular graphs with classical parameters that support a uniform structure: Case q ≥ 2

Let $\Gamma =(X,R)$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set $R$. Fix a vertex $x\in X$, and define $R_f=R\setminus \left\{yz\right|\partial (x,y)=\partial (x,z)\}$, where ∂ denotes the path-length distance in Γ. Observe that the graph ${\Gamma }_f=(X,R_f)$ is bipartite. We say that Γ supports a uniform structure with respect to x whenever ${\Gamma }_f$ has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7].

Assume that Γ is a distance-regular graph with classical parameters $(D,q,\alpha ,\beta )$ and diameter $D\ge 4$. Recall that q is an integer such that $q\notin \{-1,0\}$. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case $q\le 1$ in [3], and so in this paper we assume $q\ge 2$. Let $T=T\left(x\right)$ denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either $\alpha =0$ or $\alpha =q$, $\beta =q^2(q^D-1)/(q-1)$, and $D\equiv 0\left(\mathrm{mod}6\right)$.