Connectivity concerning the last two subconstituents of a Q-polynomial distance-regular graph

TL;DR

This paper proves that the subgraph induced on the union of the last two subconstituents of a Q-polynomial distance-regular graph, with respect to a fixed vertex, is connected.

Contribution

It establishes the connectivity of a specific subgraph related to the last two subconstituents in Q-polynomial distance-regular graphs.

Findings

The subgraph on the union of the last two subconstituents is connected.

The result applies to graphs with diameter at least 3.

Provides new insights into the structure of Q-polynomial distance-regular graphs.

Abstract

Let $\Gamma$ be a $Q$-polynomial distance-regular graph of diameter $d\geq 3$. Fix a vertex $\gamma$ of $\Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $\Gamma$ with respect to $\gamma$. We prove that this subgraph is connected.