The term “graph” in this book denotes something quite different from the graphs you may be familiar with from analytic geometry or function theory. The kind of graph you probably have dealt with consisted of the set of all points in the plane whose coordinates (x, y), in some coordinate system, satisfy an equation in x and y. The graphs we are about to study in this book are simple geometrical figures consisting of points and lines connecting some of these points; they are sometimes called “linear graphs”. It is unfortunate that two different concepts bear the same name, but this terminology is now so well established that it would be difficult to change. Similar ambiguities in the names of things appear in other mathematical fields, and unless there is danger of serious confusion, mathematicians are reluctant to alter the terminology.

The first paper on graph theory was written by the famous Swiss mathematician Euler, and appeared in 1736. From a mathematical point of view, the theory of graphs seemed rather insignificant in the beginning, since it dealt largely with entertaining puzzles. But recent developments in mathematics, and particularly in its applications, have given a strong impetus to graph theory. Already in the nineteenth century, graphs were used in such fields as electrical circuitry and molecular diagrams.