A construction of distance-regular graphs from subspaces in d-bounded distance-regular graphs

Let Gamma be a d-bounded distance-regular graph with diameter d >= 3 and with geometric parameters (d, b, alpha). Pick x is an element of V(Gamma), and let P(x) be the set of all subspaces containing x. Suppose P(x, m) is the set of all subspaces in P(x) with diameter m, where 1 <= m < d. Define a graph Gamma' whose vertex-set is P(x,m), and in which Delta(1) is adjacent to Delta(2) if and only if d(Delta(1) boolean AND Delta(2)) = m - 1. We prove that Gamma' is a distance-regular graph and compute its intersection numbers.