Let \(\Gamma \) denote a distance-regular graph with diameter \(D \ge 2\) . 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.
