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

Let Γ = (X , R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R . Fix a vertex x ∈ X , and define R f = R \ { y z ∣ ∂ (x , y) = ∂ (x , z) } , where ∂ denotes the path-length distance in Γ . Observe that the graph Γ f = (X , R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x. Assume that Γ is a distance-regular graph with classical parameters (D , q , α , β) 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 Γ 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. [ABSTRACT FROM AUTHOR]