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

Let ΓΓ be a d  -bounded distance-regular graph with diameter d⩾3d⩾3. Suppose that P(x)P(x) is a set of all strongly closed subgraphs containing x   and that P(x,i)P(x,i) is a subset of P(x)P(x) consisting of all elements of P(x)P(x) with diameter i  . Let L′(x,i)L′(x,i) be the set generated by all joins of the elements in P(x,i)P(x,i). By ordering L′(x,i)L′(x,i) by inclusion or reverse inclusion, L′(x,i)L′(x,i) is denoted by LO′(x,i) or LR′(x,i). We prove that LO′(x,i) and LR′(x,i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of LO′(x,i).