Quantum graphs: an introduction and a brief survey

The purpose of this text is to set up a few basic notions concerning quantum graphs, to indicate some areas addressed in the quantum graph research, and to provide some pointers to the literature. The paper serves as a survey, a literature guide, and an introduction that could be useful for reading other articles devoted to quantum graphs and their applications. Quantum graphs, considered as one-dimensional singular varieties equipped with differential operators, arise as simplified models in various fields such as mathematics, physics, chemistry, and engineering, particularly when studying the propagation of waves through quasi-one-dimensional systems. The paper discusses the manifold reasons for studying quantum graphs, referencing applications in areas like free-electron theory in chemistry, quantum wires, dynamical systems, and photonic crystals. It aims to guide readers to existing surveys and collections related to quantum graphs as well as critical research topics in the field.