A new approach to the Bipartite Fundamental Bound

We consider a bipartite distance-regular graph Γ with vertex set X , diameter D ≥ 4 , valency k ≥ 3 , and eigenvalues θ 0 > θ 1 > ⋯ > θ D . 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 th row and 0 in all other rows. Fix x , y ∈ X with ∂ ( x , y ) = 2 , where ∂ denotes the path-length distance. For 0 ≤ i , j ≤ D , we define w i j = ∑ z ˆ , where the sum is over all vertices z such that ∂ ( x , z ) = i and ∂ ( y , z ) = j . Define a parameter Δ in terms of the intersection numbers by Δ = ( b 1 − 1 ) ( c 3 − 1 ) − ( c 2 − 1 ) p 22 2 . In [M. MacLean, An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216], we defined what it means for Γ to be taut. We show Γ is taut if and only if Δ ≠ 0 and the vectors E x ˆ , E y ˆ , E w 11 , E w 22 are linearly dependent for E ∈ { E 1 , E d } , where d = ⌊ D / 2 ⌋ and E i is the primitive idempotent associated with θ i .