On bipartite Q-polynomial distance-regular graphs

Štefko Miklavič
2007 European journal of combinatorics (Print)  
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 , letẑ denote the vector in R X with a 1 in the z-coordinate, and 0 in all other 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{mw | m ∈ M, w ∈ W },
more » ... re 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 3d − 3 if Γ is 2-homogeneous, 3d − 1 if Γ is the antipodal quotient of the 2d-cube, and 4d − 4 otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the Q-polynomial property.
doi:10.1016/j.ejc.2005.09.003 fatcat:7ge2yrvpwbbnnbzn2sqpcpnvl4