On bipartite -polynomial distance-regular graphs

Let Γ denote a bipartite Q -polynomial distance-regular graph with vertex set X , diameter d ≥ 3 and valency k ≥ 3 . Let R X denote the vector space over R consisting of column vectors with entries in R and rows indexed by X . For z ∈ X , let z ˆ denote the vector in R X with a 1 in the z -coordinate, and 0 in all other coordinates. Fix x , y ∈ X such that ∂ ( x , y ) = 2 , where ∂ denotes the path-length distance. For 0 ≤ i , j ≤ d define w i j = ∑ z ˆ , where the sum is over all z ∈ X such that ∂ ( x , z ) = i and ∂ ( y , z ) = j . We define W = span { w i j ∣ 0 ≤ i , j ≤ d } . In this paper we consider the space M W = span { m w ∣ m ∈ M , w ∈ W } , where M is the Bose–Mesner algebra of Γ . We observe that M W is the minimal A -invariant subspace of R X which contains W , where A is the adjacency matrix of Γ . We display a basis for M W that is orthogonal with respect to the dot product. We give the action of A on this basis. We show that the dimension of M W is 3 d − 3 if Γ is 2-homogeneous, 3 d − 1 if Γ is the antipodal quotient of the 2 d -cube, and 4 d − 4 otherwise. We obtain our main result using Terwilliger’s “balanced set” characterization of the Q -polynomial property.