Geometric parameter sequences for $Q$-polynomial distance-regular graphs

Source abstract

Let $Γ$ be a distance-regular graph with diameter $D \geq 3$, and let $θ$ be a nonzero eigenvalue of $Γ$ distinct from its valency. Motivated by the geometric parameter sequences of geometric distance-regular graphs, we introduce analogous sequences defined recursively from the intersection numbers of $Γ$ and $θ$, without assuming a geometric structure. We show that the recursion is well defined if and only if all terms of the standard sequence corresponding to $θ$ are nonzero and consecutive terms are distinct. In particular, the geometric parameter sequences associated with the smallest eigenvalue always exist. Suppose that $Γ$ is $Q$-polynomial with respect to an eigenvalue ordering $θ_0,θ_1, \ldots,θ_D$. Let $\{\varphi_i\}_{i=1}^{D}$ and $\{φ_i\}_{i=1}^{D}$ be the first and second split sequences of the primary Leonard system, respectively, and let $β$ denote its fundamental parameter. We prove that the geometric parameter sequences associated with $θ_D$ exist if and only if $\varphi_i\neqφ_i$ for every $1\leq i\leq D$, and if and only if $β\neq-2$ or $D$ is even. When the sequences exist, we obtain closed formulas in terms of the primary parameter array. We introduce the split ratio sequence $ρ_i=\varphi_i/φ_i$. Using this sequence, we obtain a bidiagonal factorization of $B-θ_D I$, where $B$ is the intersection matrix of $Γ$. This factorization remains valid even when the geometric parameter sequences associated with $θ_D$ do not exist. If $Γ$ is geometric and $θ_D$ is its smallest eigenvalue, then the geometric parameter sequences associated with $θ_D$ coincide with those of $Γ$.