Horst Sachs

Horst Sachs (born March 27, 1927 in Magdeburg ; † April 25, 2016 ) was a German mathematician who mainly dealt with graph theory.

Horst Sachs 1974

Life

Sachs received his doctorate in 1958 from the Martin Luther University Halle-Wittenberg under Herbert Grötzsch ( contributions to the theory of certain isoperimetric problems ). He had been a professor at the TU Ilmenau since 1963 , where he also retired.

Sachs is best known for his contributions to the theory of the spectra of graphs , on which he wrote a monograph with others. This theory considers various matrices assigned to a graph (such as the adjacency matrix ) and examines how statements about the structure of the graph are reflected in the properties of the assigned matrices (eigenvalues, eigenvectors, characteristic polynomial). A chapter of his monograph explains the roots of this theory in applications in quantum chemistry and carries out physical applications (vibration spectrum of a membrane). Sachs also dealt with applications of graph theory in chemistry and with the history of graph theory. In 1986, Sachs re-published the book Theory of Finite and Infinite Graphs by Dénes König at Teubner.

Various theorems are named after him, including Sachs's theorem, which combines the coefficients of the characteristic polynomial of a directed graph with its structural structure from its circle subgraphs:

A is a square matrix and P (A) whose characteristic polynomial with coefficients (i = 1, ..., n) . A is understood as the adjacency matrix of a directed graph (digraph). Then by Sachs' theorem (coefficient theorem for digraphs): ${\ displaystyle a_ {i}}$ ${\ displaystyle P (A) = det (\ lambda IA) = \ lambda ^ {n} + \ lambda ^ {n-1} a_ {1} + .... + a_ {n}}$

{\ displaystyle a_ {i} = \ sum _ {L \ in {\ mathcal {L}} _ {i}} (- 1) ^ {p (L)}}

Here is the set of linear directed subgraphs of the graph with exactly i nodes. is the number of components of L, that is, the cycles of which it is composed. ${\ displaystyle {\ mathcal {L}} _ {i}}$ ${\ displaystyle p (L)}$

Hansjoachim Walther was one of his doctoral students . In 2000 Sachs received the Euler Medal with Richard A. Brualdi . Until 1974 he was chairman of the Mathematical Society of the GDR .

Fonts

with Dragos Cvetković, Michael Doob Spectra of Graphs. Theory and Applications , VEB Deutscher Verlag der Wissenschaften, Academic Press 1980, 2nd edition 1982, 3rd edition Johann Ambrosius Barth, Heidelberg 1995 (also translated into Russian in 1984)
Editor Graphs, hypergraphs and applications (Eyba Conference October 1984), Teubner 1985
Editor (on behalf of the Mathematical Society of the GDR) The development of mathematics in the GDR. On the 25th anniversary of the GDR , Deutscher Verlag der Wissenschaften, Berlin 1974
Introduction to the theory of finite graphs , Hanser 1971 and in a two-volume edition by Teubner 1970, 1972
Editor with Heinz-Jürgen Voß and Hansjoachim Walther: Contributions to graph theory (Internat. Colloquium Manebach May 1967), Teubner 1968

literature

Annette Vogt: Sachs, Horst . In: Who was who in the GDR? 5th edition. Volume 2. Ch. Links, Berlin 2010, ISBN 978-3-86153-561-4 .

Web links

John J. O'Connor, Edmund F. Robertson : Horst Sachs. In: MacTutor History of Mathematics archive .

Individual evidence

^ Mathematics Genealogy Project
↑ Horst Sachs, Michael Stiebitz 250 Years of Graph Theory. , NTM font no. Business Naturwiss., Technik, Med., Volume 24, 1987, pp. 90-94
↑ Horst Sachs, Michael Stiebitz, Robin J. Wilson Eulers Koenigsberg Letters. , Journal of Graph Theory, Vol. 12, 1988, pp. 133-139
^ Sachs relations between the circles contained in a graph and its characteristic polynomial , Pub. Math. Debrecen, Volume 11, 1964, pp. 119-134, independently proven in the same year by the electrical engineer M. Milic and the chemist L. Spialter and later independently found anew several times. See Cvetkovic, Doob, Sachs Spectra of Graphs , 1980, p. 36
↑ The degree of each node for incoming and outgoing edges is 1
^ The ICA Medals. Institute of Combinatorics and its Applications, accessed June 17, 2018 .
↑ An overview of more recent results has since appeared by Cvetkovic, Doob, Ivan Gutman, Aleksandar Torgasev Recent results in the theory of graph spectra , Annals of Discrete Mathematics, Volume 36, North Holland 1988

personal data
SURNAME	Sachs, Horst
BRIEF DESCRIPTION	German mathematician
DATE OF BIRTH	March 27, 1927
PLACE OF BIRTH	Magdeburg
DATE OF DEATH	April 25, 2016

[1] Mathematics Genealogy Project

[2] Horst Sachs, Michael Stiebitz 250 Years of Graph Theory. , NTM font no. Business Naturwiss., Technik, Med., Volume 24, 1987, pp. 90-94

[3] Horst Sachs, Michael Stiebitz, Robin J. Wilson Eulers Koenigsberg Letters. , Journal of Graph Theory, Vol. 12, 1988, pp. 133-139

[4] Sachs relations between the circles contained in a graph and its characteristic polynomial , Pub. Math. Debrecen, Volume 11, 1964, pp. 119-134, independently proven in the same year by the electrical engineer M. Milic and the chemist L. Spialter and later independently found anew several times. See Cvetkovic, Doob, Sachs Spectra of Graphs , 1980, p. 36

[5] The degree of each node for incoming and outgoing edges is 1

[6] The ICA Medals. Institute of Combinatorics and its Applications, accessed June 17, 2018 .

[7] An overview of more recent results has since appeared by Cvetkovic, Doob, Ivan Gutman, Aleksandar Torgasev Recent results in the theory of graph spectra , Annals of Discrete Mathematics, Volume 36, North Holland 1988