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

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