Parallelogram-free distance-regular graphs

Abstract

Let Gamma=(X, R) denote a distance-regular graph with distance function partial derivative and diameter d greater than or equal to 4. By a parallelogram of length i (2 less than or equal to i less than or equal to d), we mean a 4-tuple xyzu of vertices in X such that partial derivative(x, y) = partial derivative(z, u) = 1, partial derivative(x,u) = i, and partial derivative(x,z) = partial derivative(y, z) = partial derivative(y, u)=i-1. We prove the following theorem. THEOREM. Let Gamma denote a distance-regular graph with diameter d greater than or equal to 4, and intersection numbers a(1) = 0, a(2) not equal 0. Suppose Delta is Q-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. Then Gamma has classical parameters (d, b, alpha, beta) with b < -1. By including results in [3]; [9], we have the following corollary. COROLLARY. Let Gamma denote a distance-regular graph with the Q-polynomial property. Suppose the diameter d greater than or equal to 4. Then the following (i)-(ii) are equivalent. (i) Gamma contains no parallelograms of any length. (ii) One of the following (iia)-(iic) holds. (iia) Gamma is bipartite. (iib) Gamma is a generalized odd graph. (iic) Gamma has classical parameters (d b, alpha, beta) and either b < -1 or Gamma is a Hamming graph or a dual polar graph. (C) 1997 Academic Press.