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

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