Graphs and Matrices

ISBN-13: 9781848829800 / Angielski / Miękka / 2010 / 171 str.

Bapat

Opis

Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book's primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted. Further topics include those of graph theory such as regular graphs and algebraic connectivity, Laplacian eigenvalues of threshold graphs, positive definite completion problem and graph-based matrix games. Whilst this book will be invaluable to researchers in graph theory, it may also be of benefit to a wider, cross-disciplinary readership.

Recenzje

From the reviews:

"Students who have completed introductory courses in linear algebra and graph theory should be able to understand and benefit from this book. It is divided into 12 chapters. ... Each chapter includes ... a good number of references in a bibliographic format. ... A complete bibliography with all of the chapter references is available at the end of the book. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty." (J. T. Saccoman, Choice, Vol. 49 (1), September, 2011)

"The book is a study of matrices associated to graphs based on linear algebra techniques. ... The exposition is exact and clear. The proofs are presented in detail and should be understood with no difficulty by any reader with a preliminary background in linear algebra. ... Hence, the book can be used as a textbook for undergraduate level courses. Graduate students and researchers working on spectral graph theory or closely related fields will also benefit from the book." (Behruz Tayfeh-Rezaie, Mathematical Reviews, Issue 2012 f)

"A student having completed introductory courses in Linear Algebra and Graph Theory should be able to understand and benefit from this text. At the end of each of the twelve chapters there are a few exercises and a good number of references. ... this text would be a fine resource for an advanced undergraduate or someone wishing to learn more about this synergistic field of study." (John T. Saccoman, The Mathematical Association of America, June, 2011)

Spis treści

Preliminaries.- Incidence matrix.- Adjacency matrix.- Laplacian matrix.- Cycles and cuts.- Regular graphs.- Algebraic connectivity.- Distance matrix of a tree.- Resistance distance.- Laplacian eigenvalues of threshold graphs.- Positive definite completion problem.- Matrix games based on graphs.- Hints and solutions.

Blurb

Whilst it is a moot point amongst researchers, linear algebra is an important component in the study of graphs. This book illustrates the elegance and power of matrix techniques in the study of graphs by means of several results, both classical and recent. The emphasis on matrix techniques is greater than other standard references on algebraic graph theory, and the important matrices associated with graphs such as incidence, adjacency and Laplacian matrices are treated in detail.

Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph.

Such an extensive coverage of the subject area provides a welcome prompt for further exploration, and the inclusion of exercises enables practical learning throughout the book. It may also be applied to a selection of sub-disciplines within science and engineering.

Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory who want to be acquainted with matrix theoretic ideas used in graph theory, it will also benefit a wider, cross-disciplinary readership.