On a certain class of 1-thin distance-regular graphs

Abstract

Let Γ denote a non-bipartite distance-regular graph with vertex set X, diameter D ≥ 3, and valency k ≥ 3. Fix x ∈ X and let T = T(x) denote the Terwilliger algebra of Γ with respect to x. For any z ∈ X and for 0 ≤ i ≤ D, let Γ_i(z) = {w ∈ X : ∂(z, w) = i}. For y ∈ Γ₁(x), abbreviate D_jⁱ = D_jⁱ(x, y) = Γ_i(x) ∩ Γ_j(y) (0 ≤ i, j ≤ D). For 1 ≤ i ≤ D and for a given y, we define maps H_i: D_iⁱ → ℤ and V_i: D_i − 1ⁱ ∪ D_i^i − 1 → ℤ as follows:
H_i(z) = |Γ₁(z) ∩ D_i − 1^i − 1|,  V_i(z) = |Γ₁(z) ∩ D_i − 1^i − 1|.
We assume that for every y ∈ Γ₁(x) and for 2 ≤ i ≤ D, the corresponding maps H_i and V_i are constant, and that these constants do not depend on the choice of y. We further assume that the constant value of H_i is nonzero for 2 ≤ i ≤ D. We show that every irreducible T-module of endpoint 1 is thin. Furthermore, we show Γ has exactly three irreducible T-modules of endpoint 1, up to isomorphism, if and only if three certain combinatorial conditions hold. As examples, we show that the Johnson graphs J(n, m) where n ≥ 7,  3 ≤ m < n/2 satisfy all of these conditions.