# Robert Wertheimer Frucht

### Born: 9 August 1906 in Brünn, Austria-Hungary, now Brno, Czech Republic

Died: 26 June 1997 in Valparaiso, Chile

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**Roberto Frucht**was known as Robert Frucht until he went to South America in 1939 when he changed his name to Roberto Frucht. Roberto's father, Rudolf Frucht, was an editor who had Jewish relations. In 1908 the family moved to Berlin and it was in that city that Frucht was brought up. His education began at the Gymnasium Askanische in Berlin in 1912. The Askanische Gymnasium, at 24 to 26 Hallesche Strasse in the district Tempelhof district of Berlin, had been founded in 1875 and its first headmaster was Woldemar Ribbeck (1830-1902). When Frucht entered the school the director was Adolf Busse (1856-1942). Frucht passed his matriculation examination at Michaelmas 1924 (in September) giving him the right to study at university.

While he was studying at school, he became fascinated by the new developments in physics. The theory of relativity and quantum mechanics were, at this time, changing the nature of physics and the young Frucht wanted to be part of this revolution but was unsure whether reading mathematics or physics at university would be the right route to follow. He was still undecided when he entered the University of Berlin in 1924.

Taking courses in mathematics, physics and philosophy during his first few semesters at university convinced Frucht that physics was not for him. It was not the theoretical side which put him off since he enjoyed this aspect of the topic but rather it was the experimental side. Frucht quickly realised that he lacked the necessary manual dexterity to be a successful experimental physicist so came more and more to concentrate on mathematics. However, it was the mathematics relevant to the theory of relativity, namely the tensor calculus and its applications to higher dimensional differential geometry, that was his main interest. When he was looking for someone who could be his thesis advisor on this topic he realised that this was not going to be possible at the University of Berlin. He therefore went to Issai Schur, a lecturer who he greatly admired for his outstanding lecturing style, and asked him if he would act as his thesis advisor.

Schur was happy to be Frucht's thesis advisor but only on the condition that he undertook research on group theory. He completed work on his thesis

*Über die Darstellung endlicher Abelscher Gruppen durch Kollineationen*Ⓣ and was examined on 16 January 1930 by Issai Schur and Ludwig Bieberbach. On 14 March 1930, he submitted a paper based on his thesis to the

*Journal für die reine und angewandte Mathematik*. In his thesis Frucht lists all the professors who taught him at the University of Berlin: Ludwig Bieberbach, Georg Feigl, Heinz Hopf, Wolfgang Köhler (1887-1967), Karl Löwner, Richard von Mises, Walther Nerst (1864-1941), Max Planck, Alfred Pringsheim, Emil Rupp (1898-1979), Erhard Schmidt, Wilhelm Schlenk (1879-1943), Erwin Schrödinger, Issai Schur, Edward Spranger (1882-1963), Arthur Wehnelt (1871-1944) and Max Wertheimer (1880-1943). He was awarded a doctorate 'magna cum laude' in 1931. It is worth noting that, despite moving from physics to mathematics, Max Planck told him after examining him in physics:-

In 1930 the family home was in Dieffenbachstrasse in the Kreuzberg district of Berlin but, by this time Rudolf Frucht, Roberto's father, had lost his job and Frucht felt that he had to earn money to support the family. Since university jobs as a mathematician were scarce and since, as a Czechoslovakian citizen, he was not eligible to be a Gymnasium teacher in Germany, he took a position as an actuary in an Italian Insurance Company in Trieste, Italy. He writes [4]:-... hopefully my physics students understand physics as well as you.

The Italian typist whom Frucht married in 1932 was Maria Mercedes Bertogna Posselt; they had one child, a daughter Erica. Despite Frucht saying that this was a boring period for his mathematics, he did publish a number of papers. Some were related to his actuarial work, particularly annuities, such asMy work there had little to do with mathematics; rather I had to dictate business letters in German to an Italian typist who, although very efficient, soon lost her job - because she became my wife. ... Form a mathematical point of view, however, that period of my life was rather boring ...

*Un modo semplice di estrapolare le rendite vitalizie secondo il tasso d'interesse*Ⓣ (1934),

*L'assicurazione addizionale dell'essenzione dal pagamento dei premi vita decrescenti in caso d'invalidità totale*Ⓣ (1934) and

*Sulle relazioni che esistono fra due tipi di formule proposte per il calcolo approssimato delle rendite vitalizie*Ⓣ (1936). Other papers, however, continued his mathematical interests:

*Bestimmung der Hyperflächen mit konformer hypersphärischer Abbildung*Ⓣ (1934). Notice that he was writing actuarial papers in Italian but mathematical papers in German.

Frucht explains in [4] how he became a graph theorist:-

In fact Chapter 8 was entitled "Logik, Theories der Spiele, Gruppentheorie" Ⓣ. Section 5 of this chapter, the final section, was about Cayley diagrams for groups. One of the questions that Dénes König posed in this book concerned automorphisms of graphs and, after working on the problem for some time, Frucht was able to solve it. He began to publish papers relating to graph theory and group theory:... one day in1936I received from the Akademische Verlagsgesellschaft a catalogue containing a description of Dénes König's book 'Theorie der endlichen und unendlichen Graphen'.ⓉSince Chapter8promised applications to group theory, I immediately ordered the book, and beginning with the day it arrived I became an enthusiastic graph theorist.

*Die Gruppe des Petersenschen Graphen und der Kantensysteme der regulären Polyeder*Ⓣ (1937) and

*Herstellung von Graphen mit vorgegebener abstrakter Gruppe*Ⓣ (1938). This second paper contained the answer to König's question. In it Frucht proved what today is known as 'Frucht's Theorem', namely that given any finite group

*A*there exists a graph

*G*whose automorphism group is isomorphic to

*A*. Also in this paper, Frucht gives a 3-regular graph with 12 vertices and 18 edges which has a trivial automorphism group. This graph is today known as the 'Frucht graph'.

In 1938 Italy began to introduce racial laws, for instance Regio Decreto 17 Novembre 1938 Nr. 1728 banned books written by Jews, and preventing Jews holding public office or university appointments. He left his position as an actuary with the Italian Insurance Company in Trieste in 1938 and by early 1939, feeling that his future in Europe was uncertain and seeing that war was inevitable, he decided that he had to emigrate to allow his family to live in peace. His wife, Mercedes, had family in Argentina so they decided to move there. Frucht managed to find work as an actuary at an insurance company in Buenos Aires.

Robert Hermann Breusch had been a mathematics student at the University of Freiburg, but had spent considerable time studying in Berlin taking courses by Issai Schur and Richard von Mises. Frucht and Breusch were about the same age and they became friends at that time. Breusch was not Jewish but he was engaged to marry Kate Dreyfuss who was Jewish. Breusch emigrated to Chile in 1936 and found a position at the University Santa Maria in Valparaiso. In 1939 Breusch and his wife Kate (by this time they had married) emigrated to the United States and Breusch wrote to Frucht, by this time working in Buenos Aires, asking him if he would consider the academic position at the University Santa Maria in Valparaiso. He happily accepted the invitation and by the autumn of 1939 he was teaching elementary courses in Valparaiso. He was made dean of the Faculty of Mathematics and Physics in 1948. He continued in this position until 1968.

Frucht's abilities as a teacher are described in [5]:-

After taking up the position at the University Santa Maria in Valparaiso, Frucht published articles on group theory, on graph theory and on other mathematical topics. For example, early in his career in Chile he published the group theory papersThose of us who were his students received from him a solid and complete mathematical training. He was an outstanding teacher, methodical and extremely orderly in his presentation, and he managed this at all levels of education despite a heavy workload. With his powerful voice and his lively, frank and intelligent gaze, he displayed his expertise in a rigorous and very pleasant way. He wrote all the details on the board legibly, which helped us make excellent grades which have sustained us for years.

*Coronas of groups and their subgroups, with an application to determinants*(Spanish) (1942),

*The subgroups of the complete monomial groups of degree 2*(Spanish) (1944),

*On certain invariants of finite groups*(Spanish) (1945) but continued later to publish group theory papers such as

*Remarks on finite groups defined by generating relations*(1955) and

*Generalization of a theorem of Carmichael on direct products*(Spanish) (1960). Many of Frucht's graph theory papers involve both groups and graphs such as

*On the groups of repeated graphs*(1949),

*Graphs of degree three with a given abstract group*(1949) and (with Frank Harary)

*On the corona of two graphs*(1970). Frucht and Harary's introduction to this paper reads:-

Examples of Frucht's graph theory papers areOur object in this note is to construct a new and simple operation on two graphsG_{1}andG_{2}, called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups ofG_{1}andG_{2}.

*A one-regular graph of degree three*(1952),

*How to describe a graph*(1970), and

*A canonical representation of trivalent Hamiltonian graphs*(1970). Example of his papers on combinatorics are (with Gian-Carlo Rota)

*The Möbius function for partitions of a set*(Spanish) (1963),

*On inequalities and extreme values of symmetric functions on the sides of a triangle*(1964), (with Gian-Carlo Rota)

*Bell polynomials and partitions of abstract sets*(Spanish) (1965), and

*A combinatorial approach to the Bell polynomials and their generalizations*(1969).

In 1970 Frucht retired and was made professor emeritus. He continued to live in Valparaiso and devoted much time to mathematical research. Conditions for mathematicians in Chile became very difficult after a military coup took place in 1973. The universities lost many of their students and many foreign professors left the country. Frucht remained faithful to Chile although across the country as a whole, mathematical research almost stopped completely. Frucht however, continued to undertake research and publish his results although, we note, from around that time his publications are in English rather than Spanish. By the early 1980s research in mathematics, and science in general, began to be supported in Chile although it was not until 1990 that a democratic government ruled the country.

Allow me [EFR] to relate a personal connection with Frucht. On 14 January 1975 Donald Coxeter wrote to John Leech asking if he could help with a computer investigation of a class of groups call

*F*

^{a,b,c}. Leech forwarded Coxeter's letter to me and my colleague Colin Campbell since he knew we had expertise in this type of problem. We began to correspond with Leech in Scotland, Coxeter in Canada, Frucht in Chile and Abe Sinkov in Arizona who were all working on the problem. Coxeter wrote on one occasion:-

In fact the paper Coxeter refers to was never written. R M Foster was a leading electrical engineer who, in the 1920s, became interested in symmetrical graphs that could be used as electrical networks. For many decades he worked at this subject partly from a professional viewpoint and partly from a recreational one. At a conference held in Waterloo, Ontario, in April 1966, Foster presented a census of symmetric trivalent graphs with up to 400 vertices. In a letter Colin Campbell and I wrote to Coxeter on 7 October 1975, we made "theThe groups F0^{a,b,c}arose because some of them have Cayley diagrams which are '-symmetric' or 'faithful' graphs. I am writing a paper on such groups jointly with Foster, Frucht and Watkins. This is progressing very slowly.

*F*

^{a,b,c}conjecture" which, over thirty years later, was proved to be true. In 1978 Coxeter and Frucht presented a paper

*A new trivalent symmetrical graph with 110 vertices*to the 'Second International Conference on Combinatorial Mathematics' held in New York. Coxeter, Frucht and Powers published a book

*Zero-symmetric graphs*in 1981. These graphs are finite connected graphs of valency three for which the full automorphism group acts regularly on the vertices. Such graphs are both vertex-transitive and edge-transitive. The groups

*F*

^{a,b,c}yield many examples of graphs of this type with one generating involution and these examples are studied in the book.

The paper [5] contains a number of entertaining anecdotes relating to Frucht. For example, we learn that he smoked both inside and outside the classroom. He had a reputation as a car driver of an old Citroen car, mainly gained through jokes he made about his own driving. For example he joked that he had a Mercedes, but it was his wife not his car [5]:-

The authors of [5] comment that everyone knew him as "El Toro Frucht" but nobody was sure how he got that nickname [5]:-He continued driving until a very advanced age but the rumour was that his ability with advanced mathematics was far beyond his ability to drive.

An American visitor was entertained by Frucht and his wife Mercedes. He afterwards said [5]:-An intensive investigation of the origin of that nickname was not conclusive. Some students thought that was his name, while others attributed it to his pronunciation of the Greek letter "mu"; it may have been simply that his deep voice, his German accent and his tendency to say "eehh, eehh ..." sounded like mooing to students who were looking for a nickname.

One of his students described his personality:-There I had a wonderful experience in dynamic linguistic evolution. Roberto and Mercedes usually conversed in a mixture of Castilian, Italian, French and German but because of my presence, they added English to the list that day. None of us knew what language the next word would be in, but the three of us understood everything. An amazing but delightful experience.

Frucht received many honours. He was a founding member of the Mathematical Society of Chile which met informally in the late 1970s and became legally incorporated in 1983. Frucht was elected president of the Society. In 1976 he was made an honorary editor of theIn meetings and conversations, his arguments always had a crushing logic, which was a clear manifestation of his mental mathematical thinking. His frankness was sometimes disconcerting, but very sincere, genuine and well-intentioned. He did not hesitate to point out our mistakes, but corrected them only with the aim of improving them. And his modesty was proverbial. He had an excellent and fine sense of humour and laughed often with frank and sincere laughter.

*Journal of Graph Theory*, awarded the "Gabriela Mistral", Knight's Class, by the Ministry of Education in 1978, and elected a member of the Chilean Academy of Sciences in 1979. Volume 6, Issue 2, of the

*Journal of Graph Theory*contained articles "to honour Roberto Frucht." The first two articles in this issue are [7] and [4].

His death was announced by the American Mathematical Society [3]:-

Robert W Frucht, professor emeritus at Santa Maria University, Valparaiso, Chile, died on June26,1997. Born August9,1906, he was a member of the Society for29years.

**Article by:** *J J O'Connor* and *E F Robertson*

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