Distance-regular graphs with classical parameters that support a uniform structure: case q ≤ 1

2023/05/15 by Fernández, Blas, Maleki, Roghayeh, Miklavič, Štefko +1
#05C50 #05E99 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.08937

Abstract

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 ∖ \yz | ∂(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.

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