''Graph Theory, 1736–1936'' is a book in the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

. It focuses on the foundational documents of the field, beginning with the 1736 paper of

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

on the

Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in Prussia (n ...

and ending with the first textbook on the subject, published in 1936 by

Dénes Kőnig Dénes Kőnig (September 21, 1884 – October 19, 1944) was a Hungarian mathematician of Jewish heritage who worked in and wrote the first textbook on the field of graph theory. Biography Kőnig was born in Budapest, the son of mathematician Gyu ...

. ''Graph Theory, 1736–1936'' was edited by Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson, and published in 1976 by the

Clarendon Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...

. The

Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...

published a paperback second edition in 1986, with a corrected reprint in 1998.

Topics

''Graph Theory, 1736–1936'' contains copies, extracts, and translations of 37 original sources in graph theory, grouped into ten chapters and punctuated by commentary on their meaning and context. It begins with Euler's 1736 paper "Solutio problematis ad geometriam situs pertinentis" on the seven bridges of Königsberg (both in the original Latin and in English translation) and ending with Dénes Kőnig's book ''Theorie der endlichen und unendlichen Graphen''. The source material touches on

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

chemical graph theory Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo Hosoy ...

, the analysis of electrical circuits, and applications of graph theory in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

. Also included are background material and portraits on the mathematicians who originally developed this material. The chapters of the book organize the material into topics within graph theory, rather than being strictly chronological. The first chapter, on paths, includes maze-solving algorithms as well as Euler's work on

Euler tour In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...

s. Next, a chapter on circuits includes material on

knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again im ...

s in chess (a topic that long predates Euler),

Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

s, and the work of

Thomas Kirkman Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being primarily a churchman, he maintained an active interest in research-level mathematics, a ...

polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-con ...

s. Next follow chapters on

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not ...

s and

Cayley's formula In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the number of spanning tr ...

, chemical graph theory and

graph enumeration In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the gra ...

, and

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

Kuratowski's theorem In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdi ...

, and Euler's polyhedral formula. There are three chapters on the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sh ...

and

graph coloring In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

, a chapter on

algebraic graph theory Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph t ...

, and a final chapter on

graph factorization In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''- regular subgraph, and a ''k''-factorization partitions the edges of the gra ...

. Appendices provide a brief update on graph history since 1936, biographies of the authors of the works included in the book, and a comprehensive bibliography.

Audience and reception

Reviewer Ján Plesník names the book the first ever published on the history of graph theory, and although Hazel Perfect notes that parts of it can be difficult to read, Plesník states that it can also be used as "a self-contained introduction" to the field, and Edward Maziarz suggests its use as a textbook for graph theory courses. Perfect calls the book "fascinating ... full of information", thoroughly researched and carefully written, and Maziarz finds inspiring the ways in which it describes serious mathematics as arising from frivolous starting points.

Fernando Q. Gouvêa Fernando Quadros Gouvêa is a Brazilian number theorist and historian of mathematics who won the Lester R. Ford Award of the Mathematical Association of America (MAA) in 1995 for his exposition of Wiles's proof of Fermat's Last Theorem. He also w ...

calls it a "must-have" for anyone interested in graph theory, and Philip Peak also recommends it to anyone interested more generally in the history of mathematics.

References

{{DEFAULTSORT:Graph Theory, 1736-1936 Graph theory Books about the history of mathematics 1976 non-fiction books