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Description

Quantum graphs are a non-commutative generalization of classical graphs that have appeared in the context of operator algebras, non-commutative geometry and quantum information theory. Quantum graphs may be described mainly in two different ways. The first description uses the notion of operator systems or self-adjoint operator subspaces while the second description involves an abstract linear operator called quantum adjacency matrix. Several notions from classical graph theory have been successfully generalized to the setting of quantum graphs using these different descriptions of quantum graphs. However, many notions that have been studied in one model are not so well understood in the other, such as the notion of connectivity. In this talk, we will discuss a general definition of connectivity for quantum graphs using quantum adjacency matrices, which generalizes the classical notion and unifies existing notions in the literature. We will show that this new perspective simplifies and leads to quantum analogues of many classical results. This is based on joint work with Kristin Courtney and Mateuz Wasilewski.