A central object of this book is the discrete Laplace operator on
finite and infinite graphs. The eigenvalues of the discrete Laplace
operator have long been used in graph theory as a convenient tool
for understanding the structure of complex graphs. They can also be
used in order to estimate the rate of convergence to equilibrium of
a random walk (Markov chain) on finite graphs. For infinite graphs,
a study of the heat kernel allows to solve the type problem-a
problem of deciding whether the random walk is recurrent or
transient. This book starts with elementary properties of the
eigenvalues on finite graphs, continues with their estimates and
applications, and concludes with heat kernel estimates on infinite
graphs and their application to the type problem. The book is
suitable for beginners in the subject and accessible to
undergraduate and graduate students with a background in linear
algebra I and analysis I. It is based on a lecture course taught by
the author and includes a wide variety of exercises. The book will
help the reader to reach a level of understanding sufficient to
start pursuing research in this exciting area.
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