In this book, the name “quantum graph” refers to a graph con-
sidered as a one-dimensional simplicial complex and equipped with a
differential operator (“Hamiltonian”). Works that currently would be
classified as discussing quantum graphs have been appearing since at
least the 1930s in various areas of chemistry, physics, and mathemat-
ics. However, as a coherent and actively pursued topic, the area of
quantum graphs has experienced an explosive growth only in the last
couple of decades. There are manifold reasons for this surge. 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.
One can mention in particular the free-electron theory of conjugated
molecules, quantum wires, photonic crystals, carbon nano-structures,
thin waveguides, some problems of dynamical systems, system theory
and number theory, and many other applications that have led inde-
pendently to quantum graph models. Quantum graphs also play a role
of simplified, although still non-trivial, models for studying difficult
issues, for instance, Anderson localization and quantum chaos.
There are fruitful relations of quantum graphs with the older spec-
tral theory of “standard” (combinatorial) graphs [191, 195, 213–215,
415] and with what is sometimes called discrete geometric analysis
[682]. Quantum graphs present new non-trivial mathematical chal-
lenges, which makes them dear to a mathematician’s heart. As the
reader will see, work on quantum graph theory and applications has
brought together tools and intuition coming from graph theory, com-
binatorics, mathematical physics, PDEs, and spectral theory.
In the new millennium, these relations between the various topics
leading to quantum graphs were noticed, which has triggered a series
of interdisciplinary meetings and intensive communication and coop-
eration among researchers coming from different areas of science and
engineering. Surveys and collections of papers on quantum graphs and
related issues have started to appear (e.g., [98,121,122,126,161,353,
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