In this book, our goal is to introduce the main notions, structures,
and techniques used in quantum graph studies, as well as to provide
a brief survey of more special topics and applications. This task has
shaped the book as follows: we present in detail the basic constructions
and frequently used technical results in Chapters 1 and 2, devoted to
quantum graph operators, and Chapters 3 5, which address various
issues of the spectral theory of quantum graphs. The remaining two
chapters are of review nature and thus less detailed; in most cases the
reader will be directed to the cited literature for precise formulations
and proofs. Using graphs as models for quantum chaos is considered in
Chapter 6. Chapter 7 provides a brief survey of various generalizations
and applications. The reader will notice that the area is developing
very fast; had we tried to be more specific in this chapter, it would be
outdated by the time of publications anyway.
Our intent was to make the book accessible to graduate and ad-
vanced undergraduate students in mathematics, physics, and engineer-
Since a variety of techniques are used, for the benefit of the reader
we introduce the main notions and relevant results in graph theory,
functional analysis, and operator theory in a series of Appendices.
In order to make reading smoother, we normally do not include
references in the main text of the chapters, collecting them, as well
as additional comments, in the specially devoted last section of each
chapter. We also have not tried to make the considerations too gen-
eral. For instance, we mostly treat the second derivative operators on
quantum graphs, while considerations could be easily extended to the
more general Schr¨ odinger operators. When we do mention more gen-
eral operators, we do not look for the most general conditions on the
coefficients (potentials), settling for some reasonable conditions that
make the techniques work.
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