A new characterization of taut distance-regular graphs of odd diameter

We consider a bipartite distance-regular graph Γ with vertex set X, diameter D≥4, and valency k≥3. Let CX denote the vector space over C consisting of column vectors with rows indexed by X and entries in C. For z∈X, let zˆ denote the vector in CX with a 1 in the zth row and 0 in all other rows. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from z. Fix x,y∈X with distance ∂(x,y)=2. For 0≤i,j≤D, we define wij=∑zˆ, where the sum is over all vertices z∈Γi(x)∩Γj(y). Define a parameter Δ in terms of the intersection numbers by Δ=(b1−1)(c3−1)−(c2−1)p222. For 2≤i≤D−2 we define vectors wii+=∑|Γ1(x)∩Γ1(y)∩Γi−1(z)|zˆ, where the sum is over all vertices z∈Γi(x)∩Γi(y). We define W=span{wij,whh+|0≤i,j≤D,2≤h≤D−2}. In [M. MacLean, An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193-216], MacLean defined what it means for Γ to be taut. Assume D is odd. We show Γ is taut if and only if Δ≠0 and the subspace W is invariant under multiplication by the adjacency matrix.