The complete classification of triply-transitive strongly regular graphs

TL;DR

This paper completes the classification of triply-transitive strongly regular graphs by proving new cases involving polar space graphs, resulting in a definitive list of such graphs with exceptional symmetry properties.

Contribution

It proves that certain polar space graphs are triply-transitive, resolving the last open cases in the classification of these highly symmetric graphs.

Findings

Identifies the collinearity graph of $ ext{Q}^{-}(5,q)$ as triply-transitive.

Shows the affine polar graph $ ext{VO}^{ ext{ε}}_{2m}(2)$ is triply-transitive.

Provides a complete classification of all triply-transitive strongly regular graphs.

Abstract

This paper completes the classification of triply-transitive strongly regular graphs, a program recently initiated by Herman, Maleki, and Razafimahatratra. By proving that the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ and the affine polar graph $\mathrm{VO}^{\varepsilon}_{2m}(2)$ are triply-transitive, we resolve the final open cases in the classification. The result is a definitive list of all strongly regular graphs that exhibit this exceptional form of local symmetry, characterized by the equality $T_{0,\omega}=T_{\omega}=\widetilde{T}_{\omega}$ of their Terwilliger algebras.