Vertex-transitive quantum graphs

We define a quantum graph to be vertex-transitive if the join of its automorphism group is the maximum quantum relation on its quantum vertex set, in direct analogy with the classical case. All simple quantum graphs in $M_2(\mathbb{C})$ are vertex-transitive, but many simple quantum graphs in $M_3(\mathbb{C})$ are not vertex-transitive. We provide a complete classification of vertex-transitive quantum graphs in $M_3(\mathbb{C})$ up to isomorphism. To do this, we introduce a polynomial invariant for quantum graphs in $M_n(\mathbb{C})$, which we call the panoramic polynomial.