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 ∈ Γ 1 ( x ) , abbreviate D j i = D j i ( 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 i i → ℤ and V i : D i − 1 i ∪ D i i − 1 → ℤ as follows: H i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |, V i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |. We assume that for every y ∈ Γ 1 ( 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.