What are quantum graphs?

Abstract: Quantum graphs are an operator space generalization of classical graphs that have recently emerged in different branches of mathematics including operator algebras, non-commutative topology and quantum information theory. In this talk, I will provide a brief introduction to the different perspectives to quantum graphs and discuss my research interests in this direction.
We will discuss how classical concepts, such as graph coloring and chromatic numbers, can be generalized to the setting of quantum graphs. In particular, I will introduce the notion of a spectrum associated with a quantum graph using the idea of a quantum adjacency operator. It will be shown that many well-known bounds for chromatic number of classical graphs, such as Hoffman’s bound, also hold in the setting of quantum graphs. This is achieved using an algebraic characterization of quantum graph coloring associated with a nonlocal graph coloring game.