Motivated by quantum information theory, our notion of quantum graphs is introduced in early 2010's and has been developed in recent years by interactions with quantum information theory, operator algebra theory, quantum group theory, etc. Musto, Reutter, Verdon (2018) formulated the entrywise product of matrices (Schur product of operators on a commutative algebra C^n) in terms of string diagrams and applied it to noncommutative algebras. Since the adjacency matrix of a classical graph is a Schur idempotent matrix, they introduced quantum adjacency matrices as Schur idempotent operators on noncommutative algebras. In this talk we start from their approach, and see several equivalent definitions and related results. As my recent works, I would like to explain some basic examples of quantum graphs and a spectral approach to regular quantum graphs.

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In Student Colloquium, we do not fix the end time. Though a nominal time is shown above, we mean neither that it will be so, nor that it was so.