Graphs and Matrices

by Ravindra B. Bapat

Overview

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.

Product Details

ISBN-13:	9781848829800
Publisher:	Springer London
Publication date:	03/09/2011
Series:	Universitext
Edition description:	2011. Copublication with the Hindustan Book Agency
Pages:	171
Product dimensions:	6.10(w) x 9.40(h) x 0.40(d)

About the Author

Ravindra B. Bapat had his schooling and undergraduate education in Mumbai. He obtained B.Sc. from University of Mumbai, M.Stat. from the Indian Statistical Institute, New Delhi and Ph.D. from the University of Illinois at Chicago in 1981.

After spending one year in Northern Illinois University in DeKalb, Illinois and two years in Department of Statistics, University of Mumbai, Prof. Bapat joined the Indian Statistical Institute, New Delhi, in 1983, where he holds the position of Professor, Stat-Math Unit, the moment. He held visiting positions at various Universities in the U.S. and visited several Institutes abroad in countries including France, Holland, Canada, China and Taiwan for collaborative research and seminars.

The main areas of research interest of Prof. Bapat are nonnegative matrices, matrix inequalities, matrices in graph theory and generalized inverses. He has published more than 100 research papers in these areas in reputed national and international journals and guided three Ph.D. students. He has written books on Linear Algebra, published by Hindustan Book Agency, Springer and Cambridge University Press. He wrote a book on Mathematics for the general reader, in Marathi, which won the state government award for best literature in Science for 2004.

Prof. Bapat has been on the editorial boards of Linear and Multilinear Algebra, Electronic Journal of Linear Algebra, India Journal of Pure and Applied Mathematics and Kerala Mathematical Association Bulletin. He has been elected Fellow of the Indian Academy of Sciences, Bangalore and Indian National Science Academy, Delhi.

Prof. Bapat served as President of the Indian Mathematical Society during its centennial year 2007-2008. For the past several years he has been actively involved with the Mathematics Olympiad Program in India and served as the National Coordinator for the Program. Prof. Bapat served as Head, ISI Delhi Centre, during 2007-2011. He was awarded the J.C. Bose fellowship in 2009.

Table of Contents

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 to Selected Exercises.

Editorial Reviews

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)

From the Publisher