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

Abstract

Let Γ be a d-bounded distance-regular graph with diameter d ⩾ 3 . Suppose that P ( x ) is a set of all strongly closed subgraphs containing x and that P ( x , i ) is a subset of P ( x ) consisting of all elements of P ( x ) with diameter i. Let L ′ ( x , i ) be the set generated by all joins of the elements in P ( x , i ) . By 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 L O ′ ( x , i ) .