This book has been written to be of use to mathematicians working in algebraic (or more precisely, spectral) graph theory. It also contains material that may be of interest to graduate students dealing with the same subject area. It is primarily a theoretical book with an indication of possible applications, and so it can be used by computer scientists, chemists, physicists, biologists, electrical engineers, and other scientists who are using the theory of graph spectra in their work.
The rapid development of the theory of graph spectra has caused the appearance of various inequalities involving spectral invariants of a graph. The main purpose of this book is to expose those results along with their proofs, discussions, comparisons, examples, and exercises.We also indicate some conjectures and open problems that might provide initiatives for further research.
The book is written to be as self-contained as possible, but we assume familiarity with linear algebra, graph theory, and particularly with the basic concepts of the theory of graph spectra. For those who need some additional material, we recommend the books [58, 98, 102, 170].
The graphs considered here are finite, simple (so without loops or multiple edges), and undirected, and the spectra considered in the largest part of the book are those of the adjacency matrix, Laplacian matrix, and signless Laplacian matrix of a graph. Although the results may be exposed in different ways, say from simple to more complicated, or in parts by following their historical appearance, here we follow the concept of from general to specific, that is, whenever possible, we give a general result, idea or method, and then its consequences or particular cases. This concept is applied in many places, see for example Theorem 2.2 and its consequences, the whole of Subsection 2.1.2 or Theorem 2.19 and its consequences.
We briefly outline the content of the book. In Chapter 1 we fix the terminology and notation, introduce the matrices associated with a graph, give the necessary results, select possible applications, and give more details about the content. In this respect, the last section of this chapter can be considered as an extension of this Preface. In Chapters 2–4 we consider inequalities that include the largest, the least, and the second largest eigenvalue of the adjacency matrix of a graph, respectively.