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

Let Gamma denote a non-bipartite distance-regular graph with vertex set X, diameter D >= 3, and valency k >= 3. Fix x is an element of X and let T = T(x) denote the Terwilliger algebra of Gamma with respect to x. For any z is an element of X and for 0 <= i <= D, let Gamma(i) (z) = {w is an element of X : partial derivative(z, w) = i}. For y is an element of Gamma(1)(x), abbreviate D-j(i) = D-j(i)(x, y) = Gamma(i) (x) boolean AND Gamma(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) boolean OR D-i(i-1) -> Z as follows: H-i(z) = vertical bar Gamma(1) (z) boolean AND D-i-1(i-1)vertical bar, V-i(z) = vertical bar Gamma(1) (z) boolean AND D-i-1(i-1)vertical bar. We assume that for every y is an element of Gamma(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 Gamma 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.