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 \(T_{0,\omega } = \operatorname {Span}\left\{ E_{i,\omega }^*A_jE_{k,\omega }^*: 0\le i,j,k\le 2 \right\} \) . In addition, if \(G = \operatorname {Aut}(\Gamma )\) , then \(T_\omega \) is a subalgebra of the centralizer algebra \(\tilde{T}_\omega = \operatorname {End}_{G_\omega }\hspace{-0.1cm}\left( \mathbb {C}^\Omega \right) \) . 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 \(\operatorname {O}_{6}^-(q)\) , where q is a prime power, or the affine polar graph \(\textrm{VO}_{2m}^\varepsilon (2)\) , where \(m\ge 1\) and \(\varepsilon = \pm 1\) .
