With our project, we will push forward our understanding of the quite recently observed phenomenon of quantum symmetries of finite graphs as well as of the recent notion of quantum isomorphisms of graphs. We aim at finding and studying a first example of an asymmetric but quantum symmetric graph, a quantum version of the alternating group as well as new examples of non-isomorphic but quantum isomorphic graphs, and we will try to disprove the existence of an intermediate quantum permutation group. In order to do so, we will develop a computer algebraic tool checking commutativity relations in noncommutative algebras. Besides, we will investigate quantum graphs.
