Geometric parameter sequences for Q-polynomial distance-regular graphs
Abstract
Let Γ be a distance-regular graph with diameter D ≥ 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 θ0,θ1, …,θD. Let {φi}i=1D and {ϕi}i=1D 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 θD exist if and only if φi≠ϕi for every 1≤ i≤ D, and if and only if β≠−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 ρi=φi/ϕi. Using this sequence, we obtain a bidiagonal factorization of B−θD I, where B is the intersection matrix of Γ. This factorization remains valid even when the geometric parameter sequences associated with θD do not exist. If Γ is geometric and θD is its smallest eigenvalue, then the geometric parameter sequences associated with θD coincide with those of Γ.
Related
- Strongly regular graph, strongly polynomial sequences, and two-level polynomials
- Some Remarks on Regular Sequences and Complete Intersections
- Two Infinite Families of Regular Sequences of Power Sums in Three Variables
- Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences
- Kawamata-Viehweg and Lipman's vanishings for sequences of blow-ups at regular closed points
BibTeX
@misc{lee2026geometric,
title = {{Geometric parameter sequences for $Q$-polynomial distance-regular graphs}},
author = {Jae-Ho Lee and Jongyook Park and Ian Seong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26522},
eprint = {2610.26522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26522},
}
RIS
TY - UNPB
TI - Geometric parameter sequences for Q-polynomial distance-regular graphs
AU - Jae-Ho Lee
AU - Jongyook Park
AU - Ian Seong
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26522
UR - https://arxiv.org/abs/2610.26522
AB - Let Γ be a distance-regular graph with diameter D ≥ 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, …,θ_D. Let {φ_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 φ_i≠ϕ_i for every 1≤ i≤ D, and if and only if β≠−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=φ_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.
ER -