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

Abstract

Let $\hat{I}“$ denote a distance-regular graph with diameter $D\ge 2$. Let $E$ denote a primitive idempotent of $\hat{I}“$ with respect to which $\hat{I}“$ 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 $\hat{I}“$ with the above property. We describe these graphs from multiple points of view.

12 pages