Bounding the parameter β of a distance-regular graph with classical parameters

2024/10/30 by Chenhui Lv, Lv, Chenhui, Jack H. Koolen +1 · 2 citations
Mathematics · Computer Science · #Graph theory and applications #Finite Group Theory Research #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2410.22994

Abstract

Let Γ be a distance-regular graph with classical parameters (D, b, α, β) satisfying b≥ 2 and D≥ 3. Let r=1+b+b²+⋯+b^D-1. In 1999, K. Metsch showed that there exists a positive constant C(α,b) only depending on α and b, such that if β≥ C(α, b)r², then either Γ is a Grassmann graph or a bilinear forms graph. In this work, we show that for b≥ 2 and D≥ 3, then there exists a constant C₁(α, b) only depending on α and b, such that if β≥ C₁(α, b)r, then either Γ is a Grassmann graph, or a bilinear forms graph.

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