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 C X denote the vector space over C consisting of column vectors with rows indexed by X and entries in C . For z ∈ X , let z ˆ denote the vector in C X with a 1 in the z t h row and 0 in all other rows. For 0 ≤ i ≤ D , let Γ i ( 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 w i j = ∑ z ˆ , where the sum is over all vertices z ∈ Γ i ( x ) ∩ Γ j ( y ) . Define a parameter Δ in terms of the intersection numbers by Δ = ( b 1 − 1 ) ( c 3 − 1 ) − ( c 2 − 1 ) p 22 2 . For 2 ≤ i ≤ D − 2 we define vectors w i i + = ∑ | Γ 1 ( x ) ∩ Γ 1 ( y ) ∩ Γ i − 1 ( z ) | z ˆ , where the sum is over all vertices z ∈ Γ i ( x ) ∩ Γ i ( y ) . We define W = span { w i j , w h h + | 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.