Graphs and Matrices

1. Front Matter

   Pages i-ix

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2. Preliminaries

   Pages 1-10

3. Incidence Matrix

   Pages 11-23

4. Adjacency Matrix

   Pages 25-44

5. Laplacian Matrix

   Pages 45-55

6. Cycles and Cuts

   Pages 57-64

7. Regular Graphs

   Pages 65-80

8. Algebraic Connectivity

   Pages 81-94

9. Distance Matrix of a Tree

   Pages 95-109

10. Resistance Distance

    Pages 111-124

11. Laplacian Eigenvalues of Threshold Graphs

    Pages 125-135

12. Positive Definite Completion Problem

    Pages 137-144

13. Matrix Games Based on Graphs

    Pages 145-157

14. Back Matter

    Pages 169-171

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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.

• Linear and Multilinear Algebras
• Matrix Theory

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)

• Indian Statistical Institute, New Delhi, India

  R. B. Bapat

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