"I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude." (Gottfried Leibniz, [letter to Christiaan Huygens] 1670)
"A problem was posed to me about an island in the city of Königsberg, surrounded by a river spanned by seven bridges, and I was asked whether someone could traverse the separate bridges in a connected walk in such a way that each bridge is crossed only once. I was informed that hitherto no-one had demonstrated the possibility of doing this, or shown that it is impossible. This question is so banal, but seemed to me worthy of attention in that not geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether or not such a round trip is possible […]" (Leonard Euler, [letter to Giovanni Marinoni] 1736)
"There are seven bridges. If the problem could be reduced to numbers, why couldn’t I find a mathematical approach to solving it? It’s nothing to do with mathematics - it’s a purely logical problem, but that’s what intrigued me about it." (Leonhard Euler, [letter to Carl Leonhard Gottlieb Ehler, mayor of Danzig] 1736)
"Thus, you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others." (Leonhard Euler, [letter to Carl Leonhard Gottlieb Ehler, mayor of Danzig] 1736)
"A common objection to the use of mathematics in the social sciences is that the information available may only be qualitative, not quantitative. There are, however, several branches of mathematics that deal effectively with qualitative information. A very good example is graph theory." (John G Kemeny, "The Social Sciences Call on Mathematics", The Mathematical Sciences: A Collection of Essays, 1969)
"Graph theory, a special tool borrowed from topology, has now been used to reduce even quite complicated chemical structures to a chain of numbers so that a computer can analyze them." (George A W Boehm, "The Mathematical Sciences: A Collection of Essays", 1969)
"Despite the prevailing use of graphs as metaphors for communicating and reasoning about dependencies, the task of capturing informational dependencies by graphs is not at all trivial." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference", 1988)
"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."
"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)
"At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied." (Timothy Gowers, "The two cultures of mathematics", 2000)
"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws [...]. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)
"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)
"Infinite reasoning is likewise essential for graph theory. The field had its origins in topology, and it is still relevant there, but it has expanded extraordinarily far in other directions. Graph theory today is exploring the boundaries of finite provability first exposed by Gödel’s incompleteness theorem." (John Stillwell, "Mathematics and Its History", 2010)
"The most naive branch of combinatorics is graph theory, a subject that is visual and easily grasped, yet rich in connections with other parts of mathematics." (John Stillwell, "Mathematics and Its History", 2010)
"Graphs are among the most important abstract data structures in computer science, and the algorithms that operate on them are critical to modern life. Graphs have been shown to be powerful tools for modeling complex problems because of their simplicity and generality." (Jeremy Kepner & John Gilbert [Eds],"Graph Algorithms in the Language of Linear Algebra", 2011)
"The mathematical structure known as a graph has the valuable feature of helping us to visualize, to analyze, to generalize a situation or problem we may encounter and, in many cases, assisting us to understand it better and possibly find a solution." (Arthur Benjamin, "The fascinating world of graph theory", 2015)
"The theory of graphs is the fundamental study of relations in their purest, non-trivial form: binary connections between abstract points. And as so often in combinatorics, this simple assemblage of trivial objects results in a dazzlingly rich theory of seemingly endless depths." (Felix Reidl, "Structural Sparseness and Complex Networks", 2015)
"The important graphs are the ones where some things are not connected to some other things. When the unenlightened ones try to be profound, they draw endless verbal comparisons between this topic, and that topic, which is like this, which is like that; until their graph is fully connected and also totally useless." (Eliezer Yudkowsky)
"The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation." (J J Sylvester)
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