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

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.