Quantum symmetry of $3$-transitive graphs

Abstract

We study the quantum automorphism group of $3$-transitive graphs in this article. Those are highly symmetric graphs that were classified by Cameron and Macpherson in 1985, and we compute the quantum automorphism group of all such graphs, excluding the orthogonal graphs $\mathrm{O}^-(6,q)$ for $q>3$. We show that there is no quantum symmetry for the McLaughlin graph and the orthogonal graphs $\mathrm{O}^-(6,q)$ with $q = 2, 3$, while that the quantum automorphism group of the affine polar graphs $\mathrm{VO}^{+}(2k,2)$ and $\mathrm{VO}^{-}(2k,2)$ are monoidally equivalent to $\mathrm{PO}(n)$ and $\mathrm{PSp}(n)$, respectively. We use planar algebras to obtain our results, where the $3$-transitivity of the graphs gives bounds on the dimensions of the $2$-- and $3$-box spaces of the associated planar algebras.