English
资源说明：This book is concerned with results in graph theory in which linear algebra and
matrix theory play an important role. Although it is generally accepted that linear
algebra can be an important component in the study of graphs, traditionally, graph
theorists have remained by and large less than enthusiastic about using linear
algebra. The results discussed here are usually treated under algebraic graph theory,
as outlined in the classic books by Biggs [20] and by Godsil and Royle [39]. Our
emphasis on matrix techniques is even greater than what is found in these and
perhaps the subject matter discussed here might be termed linear algebraic graph
theory to highlight this aspect.
After recalling some matrix preliminaries in the Chap. 1, the next few chapters
outline the basic properties of some matrices associated with a graph. This is
followed by topics in graph theory such as regular graphs and algebraic connectivity.
Distance matrix of a tree and its generalized version for arbitrary graphs, the
resistance matrix, are treated in the next two chapters. The final chapters treat other
topics such as the Laplacian eigenvalues of threshold graphs, the positive definite
completion problem, and matrix games based on a graph.
We have kept the treatment at a fairly elementary level and resisted the temptation
of presenting up-to-date research work. Thus, several chapters in this book
may be viewed as an invitation to a vast area of vigorous current research. Only a
beginning is made here with the hope that it will entice the reader to explore further.
In the same vein, we often do not present the results in their full generality, but
present only a simpler version that captures the elegance of the result. Weighted
graphs are avoided, although most results presented here have weighted, and hence
more general, analogs.
The references for each chapter are listed at the end of the chapter. In addition, a
master bibliography is included. In a short note at the end of each chapter, we
indicate the primary references that we used. Often, we have given a different
treatment, as well as different proofs, of the results cited. We do not go into an
elaborate description of such differences.
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