A note on the classification of classical distance-regular graphs of negative type and the non-existence of hemisystems

TL;DR

This paper discusses the classification of certain algebraic graphs and the non-existence of hemisystems in Hermitian polar spaces, connecting previous results and highlighting unresolved issues due to an error in the literature.

Contribution

It links existing classifications of distance-regular graphs of negative type with the non-existence of hemisystems, clarifying the structure of these graphs and geometric objects.

Findings

Classified classical distance-regular graphs of negative type for diameter ≥ 4.

Proved non-existence of certain hemisystems in Hermitian polar spaces.

Identified an error in the literature affecting these classifications.

Abstract

DISCLAIMER: Due to an error in the literature, we cannot be sure that the conclusions drawn in this paper are correct. The goal of this note is to connect some interesting results in the literature on algebraic graph theory and finite geometry. In 1999, Weng gave an almost complete classification of classical distance-regular graphs of negative type with diameter at least 4. He proved that these graphs are either dual polar graphs of Hermitian polar spaces, Hermitian forms graphs, or fall into a last category. It was recently proved by Yian et al. that the latter category does not exist when the diameter equals 3, which by Weng's results proves that they do not exist for bigger diameter. Using a result of Vanhove, this proves that certain hemisystems in Hermitian polar spaces cannot exist.