On the classification of triply-transitive strongly-regular graphs

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 $\varepsilon = \pm 1$.