On Graph Theory (1975-1999)

"Graph theory is the study of sets of points that are joined by lines." (Martin Gardner, "Aha! Insight", 1978)

"Because problems of graph theory are so simple to describe, because graph theory is so closely tied to real-world applications, because graph theoretical problems are readily attacked by computer and graph-theoretical concepts are readily illustrated on the computer, and because it is not necessary to learn a great deal of sophisticated mathematics to work on graph-theoretical questions, graph theory is a natural subject to be introduced to students of all ages and all abilities." (Fred S Roberts, "New Directions in Graph Theory (With an Emphasis on the Role of Applications", [in "Quo Vadis, Graph Theory?"] 1993)

"Graph theory is often under attack, and so are its practitioners. We are accused of being shallow, knowing and using no real mathematics, and tackling problems of little interest, whose solutions are easy if not trivial. Although these criticisms are usually made by people unsympathetic to everything combinatorial, there is a grain of truth in these accusations - perhaps even more than a grain. In graph theory we do write too many papers, sometimes we do tackle problems that are too easy, and we have a tendency to become wrapped up in our circle of ideas and problems, unconcerned about the rest of mathematics. However, I am convinced that these are mostly teething problems." (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"Perhaps the greatest strength of graph theory is the abundance of natural and beautiful problems waiting to be solved. [...] Paradoxically, much of what is wrong with graph theory is due to this richness of problems. It is all too easy to find new problems based on no theory whatsoever, and to solve the first few cases by straightforward methods." (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"The four-color map theorem is an assertion about graph theory, which is the study of discrete points and the lines that connect them; each point is called a vertex and each line is called an edge." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"The idea of a planar graph, and other concepts which spring from that, bring together two areas of mathematics, graph theory (which is essentially the study of some relations) and the geometry of surfaces (which is part of topology). As often happens in mathematics, the marriage of two different disciplines results in some interesting (and sometimes rather difficult) offspring." (Victor Bryant, "Aspects of Combinatorics: S wide-ranging introduction", 1993)

"There are many beautiful results in graph theory whose proofs do not make use of sophisticated concepts and tools, but rather rely on great ingenuity. It is important to emphasize that this happens because there are no suitable tools available and not because a graph theorist should use as little mathematics as possible. We would be delighted to use any tools suitable to tackle the natural questions arising in the field." (Béla Bollobás, "The Future of Graph Theory", [in "Quo Vadis, Graph Theory?"] 1993)

"A graph is a good way to mathematically represent a physical situation in which there is a flow of something - materials, people, money, information – from one place to another." (John L Casti, "Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter", 1995)

"Graph theory is typical of much modern mathematics. Its subject matter is not traditional, and it is not a development from traditional theories. Its applications are not traditional either. […] Graph theory is not concerned with continuous quantities. It often involves counting, but in integers, not measuring using fractions. Graph theory is an example of discrete mathematics. Graphs are put together in pieces, in chunks, rather like Meccano or Lego, or a jigsaw puzzle." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Graphs are one of the unifying themes of computer science - an abstract representation that describes the organization of transportation systems, human interactions, and telecommunication networks. That so many different structures can be modeled using a single formalism is a source of great power to the educated programmer." (Steven S Skiena, "The Algorithm Design Manual", 1997)

"Among the parts of physics with close connections to graph theory is network theory. The earliest connection occurs in the work of Kirchhoff [1847] who, as well as formulating his two famous laws for electric circuits, made use of a graph theoretic argument to solve the resulting equations for a general electrical network. From these beginnings the links between graph theory and physics have strengthened over the centuries." (Charles Nash, "Topology and Physics - a Historical Essay", [in "History of Topology"] 1999)