The study of graph theory can take on many different forms. For example, in Computer Science one could study how graphs are used to represent data organization or the connectivity of the web. Or one could investigate many of the issues of computational complexity that are related to graphs. Compare this to the study of graph theory in the field of Mathematics, where the focus could be on how to classify types of graphs with specific properties. There are clearly various fields that can use the theory of graphs to solve relevant problems. In Graph Theory with Algorithms and its Applications, Santanu Saha Ray intends to provide a course text for students in computer science, applied mathematics and operations research. With this audience in mind, he claims to have placed a greater emphasis on algorithms and applications while still including the necessary theory. Only two chapters, however, really fulfill this promise: 5, on Algorithms and Graphs and 10, on Network Flows. The rest of the book seems to contain very few applications, and definitely not more than one could find in any other graph theory textbook.
I wanted to determine whether this text would be a reasonable choice for a course in graph theory. From my perspective as an undergraduate educator in the field of mathematics, I felt that the book is trying to reach too many audiences and thus did not do the most effective job speaking to any particular audience. In addition, due to the diversity in topics covered, there is not much transition between sections. I have found students tend to prefer books where the author gives a road map of where they are going, and how the various topics are related. The exposition in this book is fairly minimal; it seems that Ray’s intention was to present “just the facts,” without spending too much time on elaboration. In addition, though the book contains exercises, they are a bit sparse for a course textbook. For example, the first two chapters only contain one homework set. On the other hand, since the chapters can basically stand alone, it would be feasible to use a few chapters from this book as a supplement to a course.
Since this book has a diverse intended audience I wanted to see how it would compare to a book intended for undergraduate mathematics majors on the subject. For comparison I looked at Introduction to Graph Theory by Douglas B. West. It has a different focus, “on the understanding of the structure of graphs and on techniques to analyze problems in graph theory” (West, Preface). I found, however, that the material in West’s book seemed to be similar to the material in Ray’s book. For an undergraduate text book on the topic of graph theory, I found West’s book to be far superior. It includes a variety of exercises and a more detailed exposition, explaining how various topics are related. It also contained details that introduce students to logic and proof, which might well be a secondary goal of a course in graph theory.
Overall I think Graph Theory with Algorithms and its Applications could serve as an excellent reference and contains some interesting applications. The chapters are separate enough that it could be used as a starting point for several independent explorations on various topics in the field of graph theory. However, it would not be my personal choice for a course text book.
Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.