Graphs and Matrices (Universitext)

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. Annotation 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 Front Matter....Pages i-ix Preliminaries....Pages 1-10 Incidence Matrix....Pages 11-23 Adjacency Matrix....Pages 25-44 Laplacian Matrix....Pages 45-55 Cycles and Cuts....Pages 57-64 Regular Graphs....Pages 65-80 Algebraic Connectivity....Pages 81-94 Distance Matrix of a Tree....Pages 95-109 Resistance Distance....Pages 111-124 Laplacian Eigenvalues of Threshold Graphs....Pages 125-135 Positive Definite Completion Problem....Pages 137-144 Matrix Games Based on Graphs....Pages 145-157 Back Matter....Pages 169-171 Title 4 Preface 6 Table of Contents 8 1 Preliminaries 11 2 Incidence Matrix 21 3 Adjacency Matrix . 34 4 Laplacian Matrix 54 5 Cycles and Cuts 65 6 Regular Graphs 73 7 Algebraic Connectivity 89 8 Distance Matrix of a Tree 103 9 Resistance Distance 118 10 Laplacian Eigenvalues of Threshold Graphs 132 11 Positive Definite Completion Problem 143 12 Matrix Games Based on Graphs 151 Hints and Solutions to Selected Exercises 164 Bibliography 169 Index 173 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. --Book Jacket