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

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.