On the characterization of geometric distance-regular graphs
Chenhui Lv, Jack H. Koolen
TL;DR
This work addresses the problem of classifying geometric distance-regular graphs with smallest eigenvalue $-m$, diameter $D\ge3$, and $c_2\ge2$, as conjectured by Koolen and Bang. The authors introduce geometric parameters $(\phi_i,\tau_j)$, relate them to intersection numbers and eigenstructure, and employ the equal line set (ELS) property and the dual Pasch axiom to constrain possible graphs. They prove that when $\phi_1=\tau_2\ge2$, the graph is either a Johnson graph ($\phi_1=\tau_2=2$) or a Grassmann graph over $\mathbb{F}_{\tau_2-1}$ ($\phi_1=\tau_2\ge3$), or else its local graphs are $(\phi_1-1)$-clique extensions of a grid, with accompanying bounds such as $r=-\theta_{\min}$ and $\beta=k/(-\theta_{\min})$ enforcing structure. Overall, the results provide partial progress toward Koolen–Bang’s conjecture by linking geometric distance-regular graphs to partial geometries and classical graph families, shaping the landscape of possible configurations for fixed $m$.
Abstract
In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a function of $m$. In this paper, we obtain some partial results towards this conjecture.