**Structural Models: An Introduction in the Theory of Directed Graphs (1965), by Frank Harary, Robert Z Norman and Dorwin Cartwright.****1.1. Review by: Robert P Abelson.**

*J. Amer. Statistical Assoc.***61**(315) (1966), 875.This volume, despite a tendency towards hamminess in its attempt to popularise, is a clear, excellently written introduction to the mathematics of directed graphs. There is considerable overlap of coverage with Claude Berge's 'The Theory of Graphs and its Applications' (1962), but the present work has sufficient coherence, originality, even modest elegance to merit a spot of its own on a collector's bookshelf.

**1.2. Review by: B A Farbey.**

*Operations Research***17**(2) (1966), 202-203.This book is concerned with the structural properties of systems, such as communication systems, and a theory of directed graphs is developed to describe these properties precisely and in mathematical terms. It is written primarily for social scientists and provides all the necessary mathematical background for the non-mathematical reader. It is, however, a mathematician's book and proceeds in an orderly (but lively) fashion from one theorem to the next. ... The book is admirably clear and precise. Its general readability is due partly to the high standard of presentation of the text and diagrams, but primarily to the professional, "rolled-up shirtsleeves" style of writing. The notation (although one quails occasionally at the thought of yet another set of names for the same old ideas) is very easy to remember, especially for English-speaking readers. The book is an excellent introduction to the rapidly growing theory of graphs and should be kept, if not necessarily on the bookshelf of every operational research worker, at least within borrowing distance.

**1.3. Review by: D E Christie.**

*Amer. Math. Monthly***74**(1, Part 1) (1967), 104.The authorities who award prizes for attractive books should not miss this one. The authors have instilled a beguiling quality which invites browsing. The writing is readable, conversational even. Social scientists, for whom the book is principally intended, will find easy access to several basic and readily applicable sectors o digraph theory. The typography, design, and diagrams contribute admirably to the book's appeal and usefulness. Each chapter is launched by an appropriate, often waggish, quotation. How better to introduce "Limited reachability" than by "Ill can he rule the great, that cannot reach the small."

**1.4. Review by: D E Barton.**

*J Roy. Statistical Soc. Series A (General)***131**(1) (1968), 104-105.On the whole the book is clearly written with plenty of simple exercises, but Professor Harary's boundless enthusiasm makes rather extravagant calls on the reader's attention; he is so anxious that nothing be left out. More concentration on the main issues and the jettisoning of detail in the interests of elegance would have made a more exciting book.

**A Seminar on Graph Theory (1967), edited by Frank Harary.****2.1. Review by: B L Meek.**

*Math. Gaz.***53**(384) (1969), 200.Graph theory is a subject which can be enjoyed at all levels of mathematical sophistication. Many of its concepts, results (like the Königsberg Bridges or the Three Utilities) and even unsolved problems can be expressed in everyday terms in immediately comprehensible form. Even at the simple level of drawing pictures and looking at them there is ample scope for experiment and investigation, since the number of distinct graphs increases so rapidly even with only a few vertices. This book is ideal for the person with some mathematics who wants to find out what graph theory is like when more powerful techniques are brought to bear on it. It is also an admirable first introduction, for provided one is sufficient of a mathematician not to be afraid of some notation, much of it can almost be read as a novel.

**Graph Theory and Theoretical Physics (1967), edited by Frank Harary.****3.1. Review by: D J A Welsh.**

*Math. Gaz.***54**(390) (1970), 432-433.A highlight of this volume is the first appearance of verse from the pen of Blanche Descartes - a possible mate for the Bourbaki. Entitled "Enumerational" it draws attention to the little noticed work of Redfield, and to the reviewer at least in both literary merit and content it sets the tone for the whole book. Based on a series of lectures given in 1963 at the NATO Summer School on graph theory, this book concentrates on the relationship between graph theory and statistical mechanics and electric network theory, and in keeping with the editor's main interest six of the twelve lectures deal with enumerative problems. The first chapter entitled "Graphical enumeration problems" by F Harary is an elegant exposition of the state of enumerative graph theory, of value to both the expert and beginner.

**Graph Theory (1969), by Frank Harary.****4.1. Review by: Pamela Liebeck.**

*Math. Gaz.***55**(393) (1971), 338.The book is attractively presented, with many graded exercises and many diagrams, including ten full pages of diagrams for reference purposes in the Appendix. Professor Harary has an entertaining and witty style, and his book will appeal to a wide audience, serving as a student's text as well as a stimulating reference book for those interested in the subject from a less academic point of view.

**4.2. Review by: R J Wilson.**

*Amer. Math. Monthly***79**(8) (1972), 923-925.Here, at last, is the sort of book that graph theorists have been waiting for! For far too long, various mathematicians, too snobbish or ignorant to be able to peer beyond the realms of their own little 'pet' problems, have been content to dismiss the subject by describing it as "little more than recreational mathematics", or even "all right for economists and biologists, but not really 'proper' mathematics". In spite of the pioneering books of König (1936), Berge (1958) and Ore (1962), such comments are still occasionally heard; however, it is now generally realized that the tide has turned and graph theory is becoming universally recognized as 'respectable'. Professor Harary's book will undoubtedly accelerate this worthy cause. There are, however, some criticisms which should be made about the book. Be- cause of the way it is constructed, it tends at times to resemble an encyclopaedia rather than a text-book, and, as a result, the student who tries to pick up the subject on his own will almost certainly find the book heavy-going. ... All in all, Professor Harary has produced an excellent and worthwhile book which can be highly recommended; he has provided a valuable service, not only to graph theorists, but to mathematicians as a whole.

**4.3. Review by: D H Younger.**

*SIAM Review***14**(2) (1972), 350-351.It would be nice to have a text book on graph theory which would enable the initiate to learn about the concepts, become familiar with the proofs of major theorems, and discover what needs most to be done. Of course, the subject is now sufficiently broad that not everything could be fully covered in one volume. But one would hope that a general text on the subject would introduce such basic properties of graphs as connectivity, symmetry, planarity, genus, thickness, and chromatic number. One would hope to see important configurations such as trees, cutsets, n-factors, Hamiltonian cycles, and tournaments illustrated and their basic theory discussed. One would like to see proofs of Kuratowski's characterization of a planar graph, Menger's theorem on connectivity, Hall's and Tutte's theorems on 1-factors, and Dilworth's theorem for partial orderings. Harary's book 'Graph Theory' is a realization of such a hope. ... In general the book has a lightness of style that is attractive. Like everything touched by Harary, his personality comes through on every page.

**Graphical Enumeration (1973), by Frank Harary and Edgar M Palmer.****5.1. Review by: J W Moon.**

*SIAM Review***16**(2) (1974), 264-265.When considering graphs with certain properties, it is natural to ask how many such graphs there are. Such problems often seem hopelessly difficult; nevertheless, many such problems have been solved, at least in a formal sense, and this book gives an account of these results and the methods used to obtain them. Enumeration problems for graphs are usually, but not always, easier when the vertices are labelled than when they are not. In the labelled case a reasonably explicit expression for the answer can usually be deduced from first principles in a fairly straightforward way, if the problem can be solved at all ... The book is full of diagrams, examples and tables; many results are presented in the form of exercises. There is a bibliography of over 150 references and an index of symbols and definitions. Anyone who is at all interested in enumerating graphs should find this book useful, both as a reference and for instruction.

**Structural Models in Anthropology (1984), by Per Hage and Frank Harary.****6.1. Review by: C R Hallpike.**

*Man, New Series***20**(4) (1985), 759-760.The 'structural models' discussed in this book are those of graph theory, a branch of pure mathematics concerned with the properties of points joined by lines. In the words of the authors, 'Because it is essentially the study of relations, graph theory is eminently suited to the description and analysis of a wide range of structures that constitute a significant part of the subject matter of anthropology, as well as of the social sciences generally. We have in mind not only social networks, whose underlying graph theoretic basis is easily recognized, but a variety of social, symbolic, and cognitive structures as well. Belief and classification systems turn out to be no less graphical than communication and exchange networks'. Harary is one of the world's leading experts on graph theory and we are given an authoritative survey of graphs, signed graphs, digraphs, matrices, networks, and groups. Each of the eight chapters is a combination of pure graph theory, expounded at a level accessible to the non-mathematician, and a discussion of its ethnographic applications, which is Hage's contribution to the book. ... the book is an extremely useful introduction to graph theory that can be strongly recommended to the anthropologist, and there is no doubt that the basic method of thinking about problems which graph theory involves will prove increasingly fruitful in the analysis of human society and culture.

**6.2. Review by: Richard Scaglion.**

*American Anthropologist, New Series***87**(1) (1985), 171-172.Potential readers of this useful volume should not be misled by some of its advertising, which suggests that it is written with a mathematically unsophisticated audience in mind. The foreword states that the authors "write with the innumerate and mathematically phobic social scientist clearly in their sights, so that no previous acquaintance with graph theory is needed." The dust jacket blurb states, "The presentation is clear, precise, and readily accessible to the nonmathematical reader." While I agree that no previous knowledge of graph theory is needed, and that the presentation is clear and precise, mathematically phobic social scientists will nevertheless run for cover.

**6.3. Review by: Stephen B Seidman.**

*SIAM Review***27**(2) (1985), 301-303.The Hage and Harary book is intended to provide just such a formal linkage between anthropology and graph theory. Its goal is to provide anthropologists with a graph-theoretic language within which the "structural" models developed in anthropology can be given common expression. ... For mathematicians interested in the development of mathematical models in social science, the book is interesting and useful, but perhaps too narrowly focused. The authors' choice to restrict their treatment to graph-theoretic models in anthropology has two unfortunate consequences: the omission of "structural" models in anthropology that are not graph-theoretic but that are clearly related to the models that are discussed, and the omission of graph-theoretical models outside anthropology that are very similar to the models that are discussed. In both cases, the omissions leave the reader with a false impression of the current state of research in mathematical social science. ... [The book's] organization and classification of "structural" models in anthropology along graph-theoretical lines will be very useful in the education of anthropologists. Mathematicians with an interest both in discrete models generally and in mathematical social science in particular should find the anthropological models very interesting. Mathematical social science has suffered from a scarcity of bridges between social scientists and mathematicians. Hage and Harary have provided a very important bridge.

**6.4. Review by: Ellen Koskoff.**

*Ethnomusicology***30**(1) (1986), 167.This book provides a way to apply the analytical procedures of graph theory to anthropological data. In ethnological description, models implicit in the data are most frequently presented in ordinary language; this book provides a clear explanation and presentation of graphic structures that could be used to show various social and cognitive relationships.

**Exchange in Oceania: A Graph Theoretic Analysis (1991), by Per Hage and Frank Harary.****7.1. Review by: C A Gregory.**

*Man, New Series***27**(2) (1992), 425-426.The graph theoretic approach is a useful diagrammatic way to clarify the logical bases of exchange relations and an excellent means for exposing the confusions and contradictions in the work of others. Hage and Harary have found much evidence of woolly thinking in their review of the literature on Oceanic exchange and their corrections and clarifications will prove useful to authors, like myself, whose work they have corrected, clarified and modified. However, it is one thing to eliminate the ambiguity and contradictions in the reasoning of scholars but quite another to eliminate them in the exchange systems under study. But this is precisely what Hage and Harary do in their attempt to prove their claim that the 'structured islands' that bathe in the 'ocean of contingency' are more numerous than commonly imagined.

**7.2. Review by: Douglas R White.**

*American Anthropologist, New Series***95**(2) (1993), 497-498.Mathematics, like structuralism, deals with the variable content of culture by under-standing the relations among cultural elements. Network and graph theory are ideally suited for this task. In a virtual handbook of formal concepts and techniques for analysis of structure and for conceptualizing the full variety of uses of the concept of structure, anthropologist Hage and graph-theorist Harary provide foundations for the comprehensive study of structure and dispel some conceptual confusions haunting ethno-graphic literature. They focus on analyzing the diversity of exchange relations (trade, marriage, and kinship; ceremonial and social relations; global social structures) in Oceania. ... The authors demonstrate chapter by chapter their general thesis (p. 275): "For each area of study in anthropology that involves structure, there is a branch of graph theory that can serve as the appropriate mathematical model." Enlarging on their earlier book ('Structural Models in Anthropology', Cambridge University Press, 1983), also organised in terms of the concepts and techniques of graph theory, they demonstrate what we can learn by an explicit formulation of concepts, directed toward robust substantive application.

**7.3. Review by: Jane C Goodale.**

*Human Ecology***21**(2) (1993), 221-223.On the jacket of this volume the reader is informed that this study follows a previous one, Structural Models in Anthropology, by the same authors, in which they explored graph theory for use in model-building in social analysis. It is further indicated that this book is written for the non-mathematical reader. With this promise and with more than a passing interest in exchange systems in Oceania, I agreed to read and review the volume. Having "read" the book more than a few times, I do not feel that I have gained any new insights into Oceanic exchange. I am ready to admit that the fault is mine, for I believe that a more mathematically inclined, or educated reader may find such insight ...

A potential reader may wish to know why they should read this book. To do this I must turn to the conclusion. There the authors outline their accomplishments:

1. "Graph theory provides a rich and precise language for describing and classifying the great variety of exchange structures found in Oceania". I found the language often impossible to follow even with a careful discussion of theorems, matrixes, graphs given earlier.

2. "Graph theory provides techniques for calculating quantitative features of exchange networks". The purpose here is to equate exchange with access to resources and the resulting strategic (political) position of some geographic regions in a network, for example, Kiriwina in the Kula.

3. "The matrix methods associated with graph theory permit rapid and accurate analyses of large exchange systems".

4. "Graph theory provides models which, in the absence of historical data, can be used to simulate the behaviour of certain exchange systems". An example is the historical network of Lapita Pottery trade.

5. "Graph theory provides a means for enumerating structural forms ... [or] When mathematical thinking encounters an anthropological structure (e.g., 'atoms of kinship') it transforms it". An example here is Oceanic siblingship to Levi-Strauss' atoms.

6. "Through its binary operations graph theory provides a means for exact descriptions and concise notations of many complex exchange structures". I do not understand the difference between No. 6 and No. 1, above.

7. Graph theory provides clear and visually appealing models of the logical properties of exchange structures.