Nonexistence of a Class of Distance-regular Graphs

Abstract

Let Gamma denote a distance-regular graph with diameter D >= 3 and intersection numbers a(1) = 0, a(2) not equal 0, and c(2) = 1. We show a connection between the d-bounded property and the nonexistence of parallelograms of any length up to d + 1. Assume further that Gamma is with classical parameters (D, b, alpha, beta), Pan and Weng (2009) showed that (b, alpha, beta) = (-2, 2, ((-2)(D+1) -1)/3). Under the assumption D >= 4, we exclude this class of graphs by an application of the above connection.