Taut distance-regular graphs and the subconstituent algebra

We consider a bipartite distance-regular graph ΓΓ with diameter D⩾4D⩾4 and valency k⩾3k⩾3. Let X   denote the vertex set of ΓΓ and fix x∈Xx∈X. Let Γ22 denote the graph with vertex set X˘={y∈X|∂(x,y)=2}, and edge set R˘={yz|y,z∈X˘,∂(y,z)=2}, where ∂∂ is the path-length distance function for ΓΓ. The graph Γ22 has exactly k2k2 vertices, where k2k2 is the second valency of ΓΓ. Let η1,η2,…,ηk2η1,η2,…,ηk2 denote the eigenvalues of the adjacency matrix of Γ22; we call these the local eigenvalues of  ΓΓ. Let A   denote the adjacency matrix of ΓΓ. We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of ΓΓ and the eigenvalues of A  . Let T=T(x)T=T(x) denote the subalgebra of MatX(C)MatX(C) generated by A,E0*,E1*,…,ED*, where for 0⩽i⩽D0⩽i⩽D, Ei* represents the projection onto the iith subconstituent of ΓΓ with respect to x. We refer to T   as the subconstituent algebra (or Terwilliger algebra) of ΓΓ with respect to x. An irreducible T-module W is said to be thin   whenever dimEi*W⩽1 for 0⩽i⩽D0⩽i⩽D. By the endpoint of W   we mean min{i|Ei*W≠0}. We give a detailed description of the thin irreducible T  -modules that have endpoint 2 and dimension D-3D-3. MacLean [An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216] defined what it means for ΓΓ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T-modules.