Spectral Radius of Graphs provides a thorough overview of important
results on the spectral radius of adjacency matrix of graphs that
have appeared in the literature in the preceding ten years, most of
them with proofs, and including some previously unpublished results
of the author. The primer begins with a brief classical review, in
order to provide the reader with a foundation for the subsequent
chapters. Topics covered include spectral decomposition, the
Perron-Frobenius theorem, the Rayleigh quotient, the Weyl
inequalities, and the Interlacing theorem. From this introduction,
the book delves deeper into the properties of the principal
eigenvector; a critical subject as many of the results on the
spectral radius of graphs rely on the properties of the principal
eigenvector for their proofs. A following chapter surveys spectral
radius of special graphs, covering multipartite graphs, non-regular
graphs, planar graphs, threshold graphs, and others. Finally, the
work explores results on the structure of graphs having extreme
spectral radius in classes of graphs defined by fixing the value of
a particular, integer-valued graph invariant, such as: the
diameter, the radius, the domination number, the matching number,
the clique number, the independence number, the chromatic number or
the sequence of vertex degrees. Throughout, the text includes the
valuable addition of proofs to accompany the majority of presented
results. This enables the reader to learn tricks of the trade and
easily see if some of the techniques apply to a current research
problem, without having to spend time on searching for the original
articles. The book also contains a handful of open problems on the
topic that might provide initiative for the reader's research.
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