Bounding the parameter $β$ of a distance-regular graph with classical parameters
Chenhui Lv, Jack H. Koolen
TL;DR
The article extends Metsch’s bound on the parameter $\beta$ for distance-regular graphs with classical parameters by proving a linear-in-$r$ threshold that enforces geometric structure or forces the graph to belong to one of several well-known families. It develops a cohesive framework based on partial linear spaces (PLS/SPLS), the Delsarte bound, and the ELS property to derive tight structural constraints, culminating in a dichotomy: either the graph is one of Johnson, Hamming/Doob, halved cube, Gosset, Grassmann, or bilinear forms graphs, or $\beta$ is bounded by an explicit max expression in terms of $r$, $b$, and $\alpha$. The method not only tightens prior results (which allowed quadratic growth in $r$) but also explains the tightness via connections to twisted Grassmann graphs. Overall, the paper provides a robust geometric-and-incidence framework for narrowing the landscape of distance-regular graphs with classical parameters and for identifying when such graphs must be one of a few classical families.
Abstract
Let $Γ$ be a distance-regular graph with classical parameters $(D, b, α, β)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(α,b)$ only depending on $α$ and $b$, such that if $β\geq C(α, b)r^2$, then either $Γ$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(α, b)$ only depending on $α$ and $b$, such that if $β\geq C_1(α, b)r$, then either $Γ$ is a Grassmann graph, or a bilinear forms graph.