Lattices generated by strongly closed subgraphs in d-bounded distance-regular graphs

Let Gamma be a d-bounded distance-regular graph with d >= 3. Suppose that P(x) is a set of strongly closed subgraphs containing x and that P(x, i) is a subset of P(x) consisting of the elements of P(x) with diameter i. Let L(x, i) be the set generated by the intersection of the elements in P(x, i). On ordering L(x, i) by inclusion or reverse inclusion, L(x, i) is denoted by L(O)(x, i) or L(R)(x, i). We prove that L(O)(x, i) and L(R)(x, i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of P(x) on ordering P(x) by inclusion or reverse inclusion. (C) 2006 Elsevier Ltd. All rights reserved.