Geodesic-transitive graphs with large diameter

TL;DR

This paper reviews the classification of finite distance-transitive graphs, highlighting that most with diameter >4 are geodesic-transitive, and explores their geometric structures and specific examples.

Contribution

It compiles known graphs, identifies the prevalence of geodesic-transitivity in large diameter cases, and extends analysis to polar Grassmann graphs.

Findings

Graphs with diameter >4 are mostly geodesic-transitive.

Examples of distance-transitive but not geodesic-transitive graphs are provided.

Explicit descriptions of geodesics in polar Grassmann graphs are given.

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.