Distance-Regular Graphs with Classical Parameters that Support a Uniform Structure: Case q ≤ 1

Let Gamma=(X,R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x is an element of X, and define R-f=R\{yz|partial derivative(x,y)=partial derivative(x,z)}, where partial derivative denotes the path-length distance in Gamma. Observe that the graph Gamma(f)=(X,R-f)is bipartite. We say that Gamma supports a uniform structure with respect to x whenever Gamma(f) has a uniform structure with respect to x. Assume that Gamma is a distance-regular graph with classical parameters (D,q,alpha,beta) with q <= 1. Recall that q is an integer, which is not equal to 0 or -1. The purpose of this paper is to study when Gamma supports a uniform structure with respect to x. The main result of the paper is a complete classification of graphs with classical parameters with q <= 1 and D >= 4 that support a uniform structure with respect to x.