Bipartite Q-polynomial distance-regular graphs and uniform posets

TL;DR

This paper explores the relationship between bipartite distance-regular graphs with Q-polynomial structures and uniform posets, identifying conditions under which these graphs induce uniform or strongly uniform poset structures.

Contribution

It establishes a connection between Q-polynomial structures in bipartite distance-regular graphs and the uniformity properties of associated posets, including classification results.

Findings

Most cases yield uniform poset structures

Most cases yield strongly uniform poset structures

Identifies exceptions for special cases

Abstract

Let $\G$ denote a bipartite distance-regular graph with vertex set $X$ and diameter $D \ge 3$. Fix $x \in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for $\G$ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\le$ on $X$ as follows. For $y,z \in X$ let $y \le z$ whenever $\partial(x,y)+\partial(y,z)=\partial(x,z)$, where $\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.