Dual bipartite \(Q\)-polynomial distance-regular graphs and dual uniform structures

Giusy Monzillo¹

¹University of Primorska - FAMNIT

The electronic journal of combinatorics, Tome 31 (2024) no. 4

Résumé

Let $\Gamma$ denote a dual bipartite $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix $x \in X$, and let $L^*$ and $R^*$ denote the corresponding dual lowering and dual raising matrix, respectively. We show that a certain linear dependency among $R^* L^{* 2}, L^* R^* L^*, L^{* 2} R^*, L^*$ holds, and determine whether this linear dependency endow $\Gamma$ with a dual uniform or dual strongly uniform structure. Precisely, except for two special cases a dual uniform structure is always attained, and except for four special cases a dual strongly uniform structure is always attained.

Détail

DOI : 10.37236/13054

Classification : 05E30, 05C50
Mots-clés : \(Q\)-polynomial property, dual lowering matrix, dual raising matrix