A “quantum graph” is a graph considered as a one-dimensional complex and equipped with a differential operator (“Hamiltonian”). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., “meso-” or “nano-scale”) system that looks like a thin neighborhood of a graph. Since at least the 1930S, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications<

**Gregory Berkolaiko** is Professor at the Department of Mathematics, Texas A&M University, College Station, USA.

**Peter Kuchment** is University Distinguished Professor at the Department of Mathematics, Texas A&M University, College Station, USA.

Preface

Introduction

Chapter 1. Operators on Graphs

Chapter 2. Quantum Graph Operators

Chapter 3. Spectra of Quantum Graphs

Chapter 4. Spectra of Periodic Graphs

Chapter 5. Spectra of Quantum Graphs

Chapter 6. Quantum Chaos on Graphs

Chapter 7. Some Applications and Generalizations

Appendix A. Some Notions of Graph Theory

Appendix B. Linear Operators and Operator-Functions

Appendix C. Structure of Spectra

Appendix D. Symplectic Geometry and Extension Theory

Bibliography

Index