A note on the classification of classical distance-regular graphs of negative type and the non-existence of hemisystems
Sam Adriaensen, Jan De Beule, Jozefien D'haeseleer, Sam Mattheus
TL;DR
This note connects Weng's almost complete classification of classical distance-regular graphs of negative type with recent results of Tian et al. and Vanhove to derive nonexistence statements for higher-diameter cases and hemisystems in Hermitian polar spaces. By reducing to the case $d=3$ and using the known classifications, it shows that the third family in Weng's theorem cannot occur, and, via Vanhove's hemisystem-to-distance-regular-graph link, excludes hemisystems for $d>2$ when $q\neq 3$. The work thus links finite geometry and algebraic graph theory to constrain possible structures, clarifying which classical distance-regular graphs can occur and ruling out many hemisystem configurations except in the unresolved $q=3$ cases.
Abstract
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.