Vertex subsets with minimal width and dual width in $Q$-polynomial distance-regular graphs

TL;DR

This paper investigates vertex subsets called descendents in $Q$-polynomial distance-regular graphs, characterizing their properties and classifying them across classical families and infinite families with unbounded diameter.

Contribution

It extends the classification of descendents to all known infinite families with classical parameters and unbounded diameter, linking them to poset structures.

Findings

Nontrivial descendents with $w\,\geq 2$ are convex iff the graph has classical parameters.

Classification of descendents is extended to 15 infinite families with classical parameters.

Revisits and characterizes classical families using posets of descendents.

Abstract

We study $Q$-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width $w$ and dual width $w^*$ satisfy $w+w^*=d$, where $d$ is the diameter of the graph. We show among other results that a nontrivial descendent with $w\ge 2$ is convex precisely when the graph has classical parameters. The classification of descendents has been done for the 5 classical families of graphs associated with short regular semilattices. We revisit and characterize these families in terms of posets consisting of descendents, and extend the classification to all of the 15 known infinite families with classical parameters and with unbounded diameter.