Almost 2-homogeneous bipartite distance-regular graphs

Let Gamma = (X, R) denote a bipartite distance-regular graph with diameter d greater than or equal to 4, and fix a vertex x of Gamma. The Terwilliger algebra of Gamma with respect to x is the subalgebra T of Mat(X) (C) generated by A, E-0*, E-1*,...,E-d*, where A is the adjacency matrix of Gamma, and where E-i* denotes the projection onto the ith subconstituent of Gamma with respect to x. Let W denote an irreducible T-module. W is said to be thin whenever dim E-i* less than or equal to 1 (0 less than or equal to i less than or equal to d). The endpoint of W is min(i\ E-i*W not equal 0). It is known that a thin irreducible T-module of endpoint 2 has dimension d - 3, d - 2, or d - i. Gamma is said to be 2-homogeneous whenever for all i (1 less than or equal to i less than or equal to d - 1) and for all x, y, z is an element of X with partial derivative(x, y) = 2, partial derivative(x, z) = i, partial derivative(Y, Z) = i, the number \Gamma(1) (x) boolean AND Gamma(1) (y) boolean AND Gamma(i-1) (z)\ is independent of x, y, z. Nomura has classified the 2-homogeneous bipartite distance-regular graphs. In this paper we study a slightly weaker condition. Gamma is said to be almost 2-homogeneous whenever for all i (1 less than or equal to i less than or equal to d - 2) and for all x, y, z is an element of X with partial derivative(x, y) = 2, partial derivative(x, z) = i, partial derivative(y, z) = i, the number \Gamma(1)(x) boolean AND Gamma(1) (y) boolean AND Gamma(i-1) (z)\ is independent of x, y, z. We prove that the following are equivalent: (i) Gamma is almost 2-homogeneous; (ii) Gamma has, up to isomorphism, a unique irreducible T-module of endpoint 2 and this module is thin. Moreover, Gamma is 2-homogeneous if and only if (i) and (ii) hold and the unique irreducible T-module of endpoint 2 has dimension d - 3. (C) 2000 Academic Press.