Distance-regular graphs with classical parameters that support a uniform structure: case $q\ge 2$

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 $x$ in the sense of Miklavič and Terwilliger \cite{MikTer}. Assume that $\Gamma$ 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 $\Gamma$ supports a uniform structure with respect to $x$. We studied the case $q\le 1$ in \cite{FMMM}, and so in this paper we assume $q\ge 2$. Let $T=T(x)$ denote the Terwilliger algebra of $\Gamma$ with respect to $x$. Under an additional assumption that every irreducible $T$-module with endpoint $1$ is thin, we show that if $\Gamma$ supports a uniform structure with respect to $x$, then either $\alpha =0$ or $\alpha =q$, $\beta =q^2(q^D-1)/(q-1)$, and $D\equiv 0(\mathrm{mod}6)$.

arXiv admin note: substantial text overlap with arXiv:2305.08937