The complete classification of triply-transitive strongly regular graphs
Weicong Li, Hanlin Zou
TL;DR
The paper resolves the long-standing problem of classifying triply-transitive strongly regular graphs by finishing the verification for the two remaining infinite families: the collinearity graph of $\mathcal{Q}^{-}(5,q)$ and the affine polar graph $\mathrm{VO}_{2m}^{\varepsilon}(2)$. Using Terwilliger algebra methods and detailed orbit analysis of vertex stabilizers, the authors show the relevant dimension equalities $\dim(T_{0,\omega})=\dim(\widetilde{T}_{\omega})$, with a key block-decomposition value $D(\widetilde{T}_{\omega})=111132123$ yielding $\dim(\widetilde{T}_{\omega})=15$ for the critical cases. They combine these results with the known eight-family classification to produce a complete list: a strongly regular graph is triply-transitive iff it belongs to one of the eight families (a)–(h). The work solidifies the connection between local symmetry, Terwilliger algebra structure, and global automorphism actions, providing a definitive resolution to a central algebraic-combinatorics classification problem.
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 $T_{0,ω}=T_ω=\widetilde{T}_ω$ of their Terwilliger algebras.