On $Q$-polynomial distance-regular graphs with a linear dependency involving a $3$-clique

TL;DR

This paper classifies all $Q$-polynomial distance-regular graphs with diameter at least 2 that contain a 3-clique whose associated primitive idempotent vectors are linearly dependent, providing multiple characterizations.

Contribution

It provides a complete classification of such graphs, revealing their structure when a specific linear dependency condition involving a 3-clique is met.

Findings

Identifies all $Q$-polynomial distance-regular graphs with the given property.

Describes these graphs from multiple perspectives.

Establishes a classification theorem for the graphs with the linear dependency condition.

Abstract

Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 2$. Let $E$ denote a primitive idempotent of $\Gamma$ with respect to which $\Gamma$ is $Q$-polynomial. Assume that there exists a $3$-clique $\{x,y,z\}$ such that $E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. In this paper, we classify all the $Q$-polynomial distance-regular graphs $\Gamma$ with the above property. We describe these graphs from multiple points of view.