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

Mark MacLean, Štefko Miklavič

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 Γi(z) = {w ∈ X : ∂(z, w) = i}. For y ∈ Γ1(x), abbreviate Dji = Dji(x, y) = Γi(x) ∩ Γj(y) (0 ≤ i, j ≤ D). For 1 ≤ i ≤ D and for a given y, we define maps Hi: Dii → ℤ and Vi: Di − 1i ∪ Dii − 1 → ℤ as follows:
Hi(z) = |Γ1(z) ∩ Di − 1i − 1|,  Vi(z) = |Γ1(z) ∩ Di − 1i − 1|.
We assume that for every y ∈ Γ1(x) and for 2 ≤ i ≤ D, the corresponding maps Hi and Vi are constant, and that these constants do not depend on the choice of y. We further assume that the constant value of Hi 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.


Keywords


Distance-regular graph, Terwilliger algbra, subconstituent algebra.

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ISSN: 1855-3974

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