Quantum automorphism groups of homogeneous graphs

Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar'e series of $G$, meaning the series $f(z)=1+c_{1}z+c_2{z}^2+\mathrm{.}\mathrm{.}\mathrm{.}$ whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives $f$ for all homogeneous graphs having 8 vertices or less.