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In the past four decades, we have witnessed a steady development of graph theory and its applications which in the last five to ten years have blossomed out into a new period of intense activity. Some measure of this rapid expansion is indicated by the observation that, over a period of only one and a half years, more than 500 new papers on graph theory and its applications were published. The main reason for this accelerated interest in graph theory is its demonstrated applications. Because of their intuitive diagrammatic representation, graphs have been found extremely useful in modeling systems arising in physical science, engineering, social science, and economic problems. The fact is that any system involving a binary relation can be represented by a graph.
As a consequence of this rapid expansion, graph theory is now too extensive a subject for adequate presentation in a volume. Faced with the alternatives of writing a shallow survey of the greater part of the applications of graph theory or of giving a reasonably deep account of a relatively small part which is closely related to the engineering applications, I have chosen the latter. The five key topics that are covered in depth are : foundations of electrical network theory, the directed-graph solutions of linear algebraic equations, topological analysis of linear systems, trees and their generation, and the realization of directed graphs with prescribed degrees. Previously, these results have been found only in widely scattered and incomplete journal articles and institutional reports, some rather unreadable, others virtually unobtainable. In this book, I have tried to present a unified and detailed account of these applications.
An effort has been made to introduce the subject matter in the book as simple as possible. Thus, all unnecessary definitions are avoided in favor of a little longer statement. For example, an edge-disjoint union of circuits may be defined as a cire, but I prefer not to do so, since the list of definitions has already been too long. Since the terminology and symbolism currently in use in graph theory are far from standardized, the choice of terms is dictated by their applications in the five key areas covered in the book. Thus, the node is preferred to vertex or point, circuit to cycle, parallel edges to multiple edges, etc. As a result, one saving feature of the book is that many of the terms used have nearly
The guide light throughout the book has been mathematical precision. Thus, all the assertions are rigorously proved many of these proofs are believed to be new and novel. An attempt has been made to present the five key topics in a complete and logical fashion, to indicate the historical background, and to credit to the original contributors as far as I can determine. I have tried to present the material in a concise manner, using discussions and examples to illustrate the concepts and principles involved. The book also contains some of the personal contributions of the author that are not available elsewhere in the literature.
Basic Theory
Foundations of Electrical Network Theory
Directed-Graph Solutions of Linear Algebraic Equations
Topological Analysis of Linear Systems
Trees and Their Generation
The Realizability of Directed Graphs with Prescribed Degrees