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

Let Γ 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.