Geodesic-transitive graphs with large diameter
Combinatorics
2026-04-13 v2
Abstract
We review the nearly complete classification project for finite distance-transitive graphs and compile a list of all known graphs. Interestingly, we find that those graphs with diameter larger than 4, apart from a small finite number of exceptions, are geodesic-transitive. Their geodesics exhibit a clear (often geometric) structure. On the other hand, we provide examples of graphs that are distance-transitive but not geodesic-transitive, including two infinite families with diameter 3 and a few sporadic ones with diameter 3, 4 or 7. In the last section, we extend our investigation to polar Grassmann graphs and provide an explicit description of their geodesics.
Cite
@article{arxiv.2603.04935,
title = {Geodesic-transitive graphs with large diameter},
author = {Pei Ce Hua},
journal= {arXiv preprint arXiv:2603.04935},
year = {2026}
}