On the classification of triply-transitive strongly-regular graphs

TL;DR

This paper classifies all triply transitive strongly-regular graphs, excluding certain known examples, by analyzing their Terwilliger algebras and symmetry properties.

Contribution

It provides a complete classification of triply transitive strongly-regular graphs beyond known special cases, advancing understanding of their algebraic and combinatorial structure.

Findings

Identifies all such graphs not isomorphic to known polar or affine polar graphs.

Establishes conditions under which the Terwilliger algebra equals the centralizer algebra.

Shows the uniqueness of certain strongly-regular graphs with high symmetry.

Abstract

Let $\Gamma = (\Omega,E)$ be a strongly-regular graph with adjacency matrix $A_1$, and let $A_2$ be the adjacency matrix of its complement. For any vertex $\omega\in \Omega$, we define $E_{0,\omega}^*$ $E_{1,\omega}^*$ and $E_{2,\omega}^*$ to be respectively the diagonal matrices whose main diagonal is the row corresponding to $\omega$ in the matrices $I, A_1$, and $A_2$. The Terwilliger algebra of $\Gamma$ with respect to the vertex $\omega\in \Omega$ is the subalgebra $T_\omega = \left\langle I,A_1,A_2,E_{0,\omega}^*,E_{1,\omega}^*,E_{2,\omega}^* \right\rangle$ of the complex matrix algebra $\operatorname{M_{|\Omega|}}(\mathbb{C})$. The algebra $T_\omega$ contains the subspace $T_{0,\omega} = \operatorname{Span}\left\{ E_{i,\omega}^*A_jE_{k,\omega}^*: 0\leq i,j,k\leq 2 \right\}$. In addition, if $G = \Aut{\Gamma}$, then $T_\omega$ is a subalgebra of the centralizer algebra $\tilde{T}_\omega = \End{G_\omega}{\mathbb{C}^\Omega}$. The strongly-regular graph $\Gamma=(\Omega,E)$ is triply transitive if $\Gamma$ is vertex transitive and $T_{0,\omega} = T_\omega = \tilde{T}_\omega$, for any $\omega \in \Omega$. In this paper, we classify all triply transitive strongly-regular graphs that are not isomorphic to the collinearity graph of the polar space $O_{6}^-(q)$, where $q$ is a prime power, or the affine polar graph $\vo_{2m}^\varepsilon(2)$, where $m\geq 1$ and $\varepsilon = \pm 1$.