On the characterization of geometric distance-regular graphs

TL;DR

This paper investigates a conjecture about the structure of geometric distance-regular graphs with specific eigenvalue and diameter constraints, providing partial results towards classifying such graphs.

Contribution

It offers partial theoretical results supporting the conjecture that certain geometric distance-regular graphs are limited to known families or have bounded size.

Findings

Partial classification results for graphs with given eigenvalues and diameter

Identification of conditions under which graphs belong to known families

Boundedness results for the number of vertices in specific cases

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.