On Bipartite Distance-Regular Graphs with Intersection Numbers c _i = (q ⁱ − 1)/(q − 1)

Abstract

Let q denote an integer at least two. Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and intersection numbers c _i = (q ⁱ − 1)/(q − 1), 1 ≤ i ≤ D. Let X denote the vertex set of Γ and let ${V = \mathbb{C}^X}$ denote the vector space over ${\mathbb{C}}$ consisting of column vectors whose coordinates are indexed by X and whose entries are in ${\mathbb{C}}$. For ${z \in X}$, let ${{\hat z}}$ denote the vector in V with a 1 in the z-coordinate and 0 in all other coordinates. Fix ${x, y \in X}$ such that ∂(x, y) = 2, where ∂ denotes the path-length distance function. For 0 ≤ i, j ≤ D define ${w_{ij} = \sum {\hat z}}$, where the sum is over all ${z \in X}$ such that ∂(x, z) = i and ∂(y, z) = j. We define W = span{w _ij | 0 ≤ i, j ≤ D}. In this paper we consider the space ${MW={\rm span} \{mw \mid m \in M, w \in W\}}$, where M is the Bose–Mesner algebra of Γ. We observe that MW is the minimal A-invariant subspace of V which contains W, where A is the adjacency matrix of Γ. We give a basis for MW that is orthogonal with respect to the Hermitean dot product. We compute the square-norm of each basis vector. We compute the action of A on the basis. For the case in which Γ is the dual polar graph D _D (q) we show that the basis consists of the characteristic vectors of the orbits of the stabilizer of x and y in the automorphism group of Γ.