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This docnrsont has been epprovod for publlo releasa and salo ; its distribution is unlimited. All righis reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanic. Published simultaneously in Canada. Library of Congress Catalog Card No.

When I was a boy of 14 my father was so ignorant I could hardly stand to have the old man around. But when 1 got to be 21, I was astonished at how much the old man had learned in 7 years.

There are several reasons for the acceleration of interest in graph theory. It has become fashionable to mention that there are applications of graph theory to some areas of physics, chemistry, communication science, computer technology, electrical and civil engineering, architecture, operational research, genetics, psychology, sociology, economics, anthropology, and linguistics.

The theory is also intimately related to many branches of mathematics, including group theory, matrix theory, numerical analysis, probability, topology, and combinatorics. The fact is that graph theory serves as a mathematical model for any system involving a binary relation. Partly because of their diagrammatic representation, graphs have an intuitive and aesthetic appeal.

Although there are many results in this field of an ele- mentary nature, there is also an abundance of problems with enough combinatorial subtlety to challenge the most sophisticated mathematician. Earlier versions of this book have been used since when regular courses on graph theory and combinatorial theory began in the Department of Mathematics at the University of Michigan.

It has been found pedagogi- cally advantageous not to in:iude proofs of all theorems. This device has permitted the inclusion of more theorems than would otherwise have been possible. The book can thus be used as a text in the tradition of the "Moore Method.

Note, however, that some of the missing proofs are both difficult and long. The reader who masters the content of this book will be qualified to continue with the study of special topics and to apply graph theory iO other fields. An effort has been made to present the various topics in the theory of graphs in a logical ordei, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results.

In addition, there are three appendices which provide diagrams of giaphs. There are vast differences in the level of exercises. Those exercises which are neither easy nor straightforward are so indicated by a bold-faced number. Exercises which are really formidable are both bold faced and starred. The reader is encouraged to consider every exercise in order to become familiar with the material which it contains.

Many of the "easier" exercises may be quite difficult if the reader has not first studied the material in the chapter. The reader is warned not to get bogged down in Chapter 2 and its many exercises, which alone can be used as a miniature course in graph theory for college freshmen or high-school seniors. The instructor can select material from this book for a one-semester course on graph theory, while the entire book can serve for a one-year course. Some of the later chapters are suitable as topics for advanced seminars.

Since the elusive attribute known as "mathe- matical maturity" is really the only prerequisite for this book, it can be used as a text at the undergraduate or graduate level.

An acquaintance with elementary group theory and matrix theory would be helpful in the last four chap. I owe a substantial debt to many individuals for their invaluable as- sistance and advice in the preparation of this book.

Lowell Beineke and Gary Chartrand have been the most helpful in this respect over a period of many years! For the past year, my present doctoral students, Dennis Geller, Bennet Manvel, and Paul Stockmeyer, have been especially enthusiastic in supplying comments, suggestions, and insights. Most recently, Branko Grnbaum and Dominic Welsh kindly gave the complete book a careful reading.

I am personally responsible for all the errors and most of the off-color remarks. Over the past two decades research support for published papers in the theory of graphs was received by the author from the Air Force Office of Scientific Research, the National Institutes of Health, the National Science Foundation, the Office of Naval Research, and the Rockefeller Foundation.

During this time I have enjoyed the hospitality not only of the University of Michigan, but also of the various other scholarly organizations which 1 have had the opportunity to visit. Finally, the author is especially grateful to the Addison-Wesley Publishing Company for its patience in waiting a full decade for this manuscript from the date the contract was signed, and for its cooperation in all aspects of the production of this book.

I hate quotations. Tell me what you know. Cutpoinls, bridges, and blocks 26 Block graphs and cutpoint graphs Centers and centroids. Connectivity and line-connectivity 43 Graphical variations of Menger's theorem 47 Further variations of Menger's theorem Plane and planar graphs Outerplanar graphs Kuratowski's theorem Other characterizations of planar graphs Genus, thickness, coarseness, crossing number The adjacency matrix.

It is no coincidence that graph theory has been independently discovered many times, since it may quite properly be regarded as an area of applied mathematics. Subsequent rediscoveries of graph theory by Kirchhoff and Cayley also had their roots in the physical world.

Kirchhoff s investiga- tions of electric networks led to his development of the basic concepts and theorems concerning trees in graphs, while Cayley considered trees arising from the enumeration of organic chemical tsomers. Another puzzle approach to graphs was proposed by Hamilton. After this, the celebrated Four Color Conjecture came into prominence and has been notorious ever since. In the present century, there have already been a great many rediscoveries of graph theory which we can only mention most briefly in this chronological account.

There were two islands linked to each other and to the banks of the Pregel River by seven bridges as shown in Fig. The problem was to begin at any of the four land areas, walk across each bridge exactly once and return to the starting point.

One can easily try to. B Fig. A park in Knigsberg, In proving that the problem is unsolvable, Euler replaced each land area by a point and each bridge by a line joining the corresponding points, thereby producing a "graph.

Showing that the problem is unsolvable is equivalent to showing that the graph of Fig. Rather than treating this specific situation, Euler generalized the problem and developed a criterion for a given graph to be so traversable; namely, that it is connected and every point is incident with an even number of lines.

While the graph in Fig. Thus, in effect, Kirchhoff replaced each electrical network by its underlying graph and showed that it is not necessary to consider every cycle in the graph of an electric network separately in order to solve the system of equations. Instead, he pointed out by a simple but powerful construction, which has since become standard procedure, that the inde- pendent cycles of a graph determined by any of its "spanning trees" will suffice.

A network N, its underlying graph G, and a spanning tree T. Of course, Cayley restated the problem abstractly: find the number of trees with p points in which every point has degree 1 or 4. He did not immediately succeed in solving this and so he altered the problem until he was able to enumerate: rooted trees in which one point is distinguished from the others , trees, trees with points of degree at most 4, and finally the chemical problem of trees in which every point has degree 1 or 4, see [C3].

Jordan later independently discovered trees as a purely mathematical dis- cipline, and Sylvester wrote that Jordan did so "without having any suspicion of its bearing on modern chemical doctrine," see [K10, p. The player is challenged to travel "around the world" by finding a closed circuit along the edges which passes through each vertex exactly once. Ham: on sold his idea to a maker of games for 25 guineas; this was a shrewd move since the game was not a financial success.

In graphical terms, the object of the game is to find a spanning cycle in the graph of the dodecahedron, shown in Fig.

This remarkable problem can be explained in five minutes by any mathematician to the so- called man in the street. At the end of the explanation, both will understand the problem, but neither will be able to solve it. The following quotation from the definitive historical article by May [M5] states the Four Color Conjecture and describes its role: [The conjecture states that] any map on a plane or the surface of a sphere can he colored with only four colors so that no two adjacent countries have the same color.

Each country must consist of a single connected region, and adjacent countries are those having a boundary line not merely a single point in common. The consensus is that the con- jecture is correct but unlikely to be proved in general. It seems destined to retain for some time the distinction of being both the simplest and most fascinating unsolved problem of mathematics.

The Four Color Conjecture has an interesting history, but its origin remains somewhat vague. There have been reports that Mbius was familiar with this problem in , but it is only definite that the problem was com- municated to De Morgan by Guthrie about The first of many erroneous "proofs" of the conjecture was given in by Kempe [K6].

An error was found in by Heawood [H38] who showed, however, that the conjecture becomes true when "four" is replaced by "five. The Four Color Conjecture is a problem in graph theory because every map yields a graph in which the countries including ihe exterior region are the points, and two points are joined by a line whenever the corresponding countries are adjacent.

Such a graph obviously can be drawn in the plane without intersecting lines. Thus, if it is possible to color the points of every planar graph with four or fewer colors so that adjacent points have different colors, then the Four Color Conjecture will have been proved.

It was pointed out that Lewin was actually dealing with graphs, as indicated by Fig. This viewpoint led the psy- chologists at the Research Center for Group Dynamics to another psycho- logical interpretation of a graph, in which people are represented by points and interpersonal relations by lines.

Such relations include love, hate, communication, and power. In fact, it was precisely this approach which led the author to a personal discovery of graph theory, aided and abetted by psychologists L. Festinger and D. The world of theoretical physics discovered graph theory for its own purposes more than once.

In the study of statistical mechanics by Uhlenbeck [Ul], the points stand for molecules and two adjacent points indicate nearest neighbor interaction of some physical kind, for example, magnetic attraction or repulsion. In a similar interpretation by Lee and Yang [LYl], the points stand for small cubes in euclidean space, where each cube may or may not be occupied by a molecule.

Then two points are adjacent whenever both spaces arc occupied. Another aspect of physics employs graph theory rather as a pictorial device. Feynmann [F3] proposed the diagram in which the points represent physical particles and the lines represent paths of the particles after collisions.

The study of Markov chains in probability theory see, for example, Feller [F2. This is made explicit in the book [HNC1, p. A similar representation of a directed graph arises in that part of numerical analysis involving matrix inversion and the calculation of eigenvalues.

Examples are given by Varga [V2, p. A square matrix is given, preferably "sparse," and a directed graph is associated with it in the following way.

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## Frank Harary

Frank Harary March 11, — January 4, was an American mathematician , who specialized in graph theory. He was widely recognized as one of the "fathers" of modern graph theory. He broadened the reach of this field to include physics, psychology, sociology, and even anthropology. Gifted with a keen sense of humor, Harary challenged and entertained audiences at all levels of mathematical sophistication.

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## Graph Theory

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## Frank Harary - Graph Theory

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