Recent research has shown that homomorphism counts can serve as a key tool to understand finite graphs and groups. In this project, we will tackle homomorphism counts from a new perspective. In fact, we do so both from a classical perspective within algebraic combinatorics as well as from a quantum perspective within quantum information theory. A better understanding of the nature and emergence of homomorphism counts and quantum homomorphism counts will provide insights into symmetries and quantum symmetries of graphs, different notions of isomorphisms of graphs, classical and quantum invariants of graphs, as well as winning strategies for non-local games.
