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. Works that
currently would be classified as discussing quantum graphs have
been appearing since at least the 1930s, and since then, quantum
graphs techniques have been applied successfully in various areas
of mathematical physics, mathematics in general and its
applications. One can mention, for instance, dynamical systems
theory, control theory, quantum chaos, Anderson localization,
microelectronics, photonic crystals, physical chemistry,
nano-sciences, superconductivity theory, etc. Quantum graphs
present many non-trivial mathematical challenges, which makes them
dear to a mathematician's heart. Work on quantum graphs has brought
together tools and intuition coming from graph theory,
combinatorics, mathematical physics, PDEs, and spectral theory.
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.
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