# Frank Harary

Frank Harary (left) and Klaus Wagner in Oberwolfach, 1972

Frank Harary (March 11, 1921 – January 4, 2005) was a prolific American mathematician, who specialized in graph theory. He was widely recognized as one of the "fathers" of modern graph theory.[1] Harary was a master of clear exposition and, together with his many doctoral students, he standardized the terminology of graphs. 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. A particular trick he employed was to turn theorems into games - for instance, students would try to add red edges to a graph on six vertices in order to create a red triangle, while another group of students tried to add edges to create a blue triangle (and each edge of the graph had to be either blue or red). Because of the simplest case of Ramsey theory, one team or the other would have to win.

## Biography

Frank Harary was born in New York City, the oldest child to a family of Jewish immigrants from Syria and Morocco. He earned his Bachelor's and Master's degrees from Brooklyn College in 1941 and 1945 respectively[2] and his Ph.D. from University of California at Berkeley in 1948. During 1948-1986 he was with the University of Michigan. From 1987 he was Professor (and Distinguished Professor Emeritus) in the Computer Science Department at New Mexico State University in Las Cruces. He was one of the founders of the Journal of Combinatorial Theory and the Journal of Graph Theory.[1] He died at Memorial Medical Center in Las Cruces, New Mexico.[3]
Frank Harary's interests would shift much during his very long and diverse academic career. His interests during the beginnings of his undergrad were mainly in the field of physics, however when he began research for his doctorate his attention was turned to abstract algebra, specifically the study of Boolean-like rings. Shortly after being awarded his doctorate for his research on Boolean-like rings, Harary was appointed as an instructor for the Department of Mathematics at the University of Michigan in 1948. Harary's first publication would follow within his first year; in 1949 Harary published On the algebraic structure of knots. Shortly after this publication in 1953 Harary published his first book (jointly with George Uhlenbeck) On the number of Husimi trees. It was following this text that Harary began to build up a world wide reputation for his work in graph theory. In 1965 Harary's first book Structural models: An introduction to the theory of directed graphs was published, and For the rest of his life Harary's interest would be in the field of Graph Theory.
While beginning his work in graph theory around 1965, Harary began buying up property in Ann Arbor to supplement income for his family. Harary and his wife Jayne had six children together, Miriam, Natalie, Judith, Thomas, Joel and Chaya. In 1969 The Michigan Daily published an article discussing issues that tenants of Harary's property were facing. Harary's motivation for purchasing these properties was in the interest of the land value, and as issues arose within the properties in terms of maintaining a safe living environment negative attention was turned towards Harary. It was Harary's intention to maintain ownership of the land and see the tenants out of their apartments while helping them find better housing.[4]
From 1973 to 2007 Harary jointly wrote five more books, each in the field of graph theory. In the time before his death, Harary traveled the world researching and publishing papers (with some 300 different co-authors) which appeared in mathematical journals and other scientific publication. Harary recoded that he lectured in 166 different cities around the United States and some 274 cities in over 80 different countries. Harary was particularly proud that he had given lectures in cities around the world beginning with every letter of the alphabet, even including "X" when he traveled to Xanten, Germany.
It was in 1986 at the age of 65 that Harary retired from his professorship at the University of Michigan. Harary did not take his retirement lightly however, following his retirement Harary was appointed as a Distinguished Professor of Computer Sciences at New Mexico State University in Las Cruces. He held this position until his death in 2005. The same year as his retirement Harary was made an honorary fellow of the National Academy of Sciences of India, he also served as an editor for about 20 different journals focusing primarily on graph theory and combinatorial theory. It was following his retirement that Harary was elected to be an honorary lifetime member of the Calcutta Mathematical Society and of the South African Mathematical Society.
At the time of his death in Las Cruces other members of the department of Computer Science felt the loss for the great mind that once worked beside them. The head of the department of Computer Science at the time of Harary's death Desh Ranjan had this to say, "Dr. Harary was a true scholar with a genuine love for graph theory which was an endless source of new discoveries, beauty, curiosity, surprises and joy for him till the very end of his life."[4]

## Mathematics

Harary's work in graph theory was diverse. Some topics of great interest to him were:

• Graph enumeration, that is, counting graphs of a specified kind. He coauthored a book on the subject (Harary and Palmer 1973). The main difficulty is that two graphs that are isomorphic should not be counted twice; thus, one has to apply Pólya's theory of counting under group action. Harary was an expert in this.
• Signed graphs. Harary invented this branch of graph theory,[5] which grew out of a problem of theoretical social psychology investigated by the psychologist Dorwin Cartwright and Harary.[6]
• Applications of graph theory in numerous areas, especially to social science. He was co-author of John Wiley's first eBook, Graph Theory and Geography.

Among over 700 scholarly articles Harary wrote, two were co-authored with Paul Erdős, giving Harary an Erdős number of 1.[7] He lectured extensively and kept alphabetical lists of the cities where he spoke.

Harary's first publication Atomic Boolean-like rings with finite radical went through much effort to be put into the Duke Mathematical Journal in 1950. This article was first submitted to the American Mathematical Society in November 1948, then sent to the Duke Mathematical Journal where it was revised three times before it was finally published two years after its initial submission. Harary began his teaching career at the University of Michigan in 1953 where he was first an assistant professor, then associate professor in 1959 and finally was appointed as a professor of mathematics in 1964. Prior to beginning his teaching career he became a research assistant in the Institute of Social Research at the University of Michigan.

Harary's most famous classic book Graph Theory was published in 1969 and offered a practical introduction to the field of graph theory. It is evident that Harary's focus in this book and amongst his other publications was towards the varied and diverse application of graph theory to other fields of mathematics, physics and many others. Taken from the preface of Graph Theory, Harary notes...

"...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."[8]

Harary quickly began promoting inquiry based learning through his texts, apparent by his reference to the tradition of the Moore Method. Harary made many unique contributions to graph theory as he explored more and more different fields of study and successfully attempted to relate them to graph theory. Harary's classic book Graph Theory begins by providing the reader with much of the requisite knowledge of basic graphs and then dives right into proving the diversity of content that is held within graph theory. Some of the other mathematical fields that Harary directly relates to graph theory in his book begin to appear around chapter 13, these topics include Linear Algebra, and Abstract Algebra.

## Key Theorem in Graph Theory

Much of Harary's work was done to further research in the field of graph theory and then almost immediately apply the tools provided by graph theory to other mathematical subjects. One such example of Harary's efforts to create this connected network between the mathematical subjects of graph theory and linear algebra was his wok on The Square of a Tree. In order to outline the connection here Harary observed the adjacency matrix for many different types of connected graph and noticed a connection between the square root of an adjacency matrix and the graph of some tree. For example given a complete graph where all entries of the adjacency matrix are 1's, it is possible to find a unique tree with which you can square it's adjacency matrix to obtain the adjacency matrix for the initial complete graph. Here we will provide a basic breakdown of this theorem, and the method used by Harary to find the unique "tree square root" of particular connected graphs.

$*$ Observe the (square) adjacency matrix A, of a graph G where A = (aij) where aij = 1 if points bi and bj are adjacent and aij=0 otherwise, except that we take the diagonal of A to contain only 1's.
$*$ If a graph G is the square of a tree, then it has a unique tree square root
$*$ Some vocabulary necessary to understand this proof and the methods used here are provided in Harary's The Square of a Tree[9]

(Complete graph, clique, cliqual, unicliqual, multicliqual, cocliqual, neighborhood, neighborly, cut point, block)

-How to determine if some graph G is the square of a tree.

-Iff a graph G is complete or satisfies the following 5 properties then G = T2
(i) Every point of G is neighborly and G is connected.
(ii) If two cliques meet at only one point b, then there is a third clique with which they share b and exactly one other point.
(iii) There is a 1-1 correspondence between the cliques and the multicliqual points b of G such that clique C(b) corresponding
to b contains exactly as many multicliqual points as the number of cliques which include b.
(iv) No two cliques intersect in more than two points.
(v) The number of pairs of cliques that meet in two points is one less than the number of cliques.


-Algorithm for finding the tree square root of a graph G.

-Step 1: Find all the cliques of G.
-Step 2: Let the cliques of G be C1,...,Cn, and consider a collection of multicliqual points
b1,...,bn corresponding to these cliques in accordance with condition iii.  The elements of this collection are
the nonendpoints of T.  Find all of the pairwise intersections of the n cliques and form the graph S by joining the points
bi and bj by a line if and only if the corresponding cliques Ci and Cj intersect in
two points.  S is then a tree by condition v.
-Step 3: For each clique Ci of G, let ni be the number of unicliqual points.  To the tree S obtained in step 2,
attach ni endpoints to bi, obtaining the tree T which we sought.


Once we have the tree in question we can create an adjacency matrix for the tree T and check that it is indeed to correct tree which we sought. Squaring the adjacency matrix of T should yield an adjacency matrix for a graph which is isomorphic to the graph G which we started with. Probably the simplest way to observe this theorem in action is to observe the case which Harary mentions in The Square of a Tree.[9] in which Harary observes this theorem in action with connected graphs. Specifically the example in question describes the tree corresponding the graph of K5

"Consider the tree consisting of one point joined with all the others. When the tree is squared, the result is the complete graph. We wish to illustrate... T2$=$K5"

Upon squaring of the adjacency matrix of the previously mentioned tree, we can observe that this theorem does in fact hold true. We can also observe that this pattern of setting up a tree where "one point joined with all the others" will always indeed yield the correct tree for all complete graphs.

## Bibliography

• Harary, Frank, Robert Z. Norman, and Dorwin Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs. New York: Wiley, 1965.
• Harary, Frank, and Edgar M. Palmer (1973), Graphical Enumeration. Academic Press, New York, NY.
• Arlinghaus, Sandra Lach, William C. Arlinghaus, and Frank Harary (2002), Graph Theory and Geography: An Interactive E-Book. New York: John Wiley and Sons.
• Hage, Per and Harary, Frank (1991), Exchange in Oceania: A Graph Theoretic Analysis (Oxford Studies in Social and Cultural Anthropology) , Oxford University Press.
• Hage, Per and Harary, Frank (1984), Structural Models in Anthropology (Cambridge Studies in Social and Cultural Anthropology), Cambridge University Press.
• Hage, Per and Harary, Frank (2007), Island Networks: Communication, Kinship, and Classification Structures in Oceania (Structural Analysis in the Social Sciences), Cambridge University Press.
• Harary, Frank (Editor) (1973), New Directions in the Theory of Graphs: Proceedings of the 1971 Ann Arbor Conference on Graph Theory, University of Michigan, Academic Press.
• Buckley, Fred and Harary, Frank (1990), Distance in Graphs, Perseus Press.
• Wilf, Herbert S. and Harary, Frank (Editors) (1971), Mathematical Aspects of Electrical Networks Analysis (Siam-Ams Proceedings, Volume 3), Symposium in Applied Mathematics, American Mathematical Society.
• Harary, Frank (Editor) (1979), Topics in Graph Theory, New York Academy of Sciences.
• Harary, Frank (1967), Graph Theory and Theoretical Physics, Academic Press.