The concept of a graph

Graph Theory is an important part of mathematics with many applications in Computer Science, Biology and Economics. Graphs are also interesting in and of themselves. Sometimes just drawing a picture is enough to solve the problem.

Problem 1. Cosmic liaisons are established among the nine planets of the solar system. Rockets travel along the following routes: Earth-Mercury, Pluto-Venus, Earth-Pluto, Pluto-Mercury, Mercury-Venus, Uranus-Neptune, Neptune-Saturn, Saturn-Jupiter, Jupiter-Mars, Mars-Uranus. Can a traveler get from Earth to Mars?

Solution. We can draw a diagram, in which the planets will be presented by points, and the routes connecting them by non-intersecting line segments, see Figure 1

  

Figure 1: The planets as a graph

It is now clear that it is impossible to travel from Earth to Mars.

Such a diagram is called a graph. The points are called the vertices of the graph, and the lines are called its edges

Problem 2. In a country there are nine cities, with the names 1, 2, 3, 4, 5, 6, 7, 8, 9. A traveler finds that two cities are connected by an airplane route if and only if the two-digit number formed by naming one city, then the other, is divisible by 3. Can the traveler get from City 1 to City 9?

Solution. If the number AB is divisible by 3, then so is the number BA. This means that if a traveler can get from city A to city B directly, he can also get from city B directly to city A. This observation allows us to draw a graph of the connections. Clearly a traveler cannot get from city 1 to city 9. Try to make an appropriate picture describing the problem.

The solution of these two problems, which do not resemble each other on the surface, have a central idea in common: the representation of the problem by a diagram, i.e. graph. An accurate definition of a graph would be too complicated. If you are a teacher, the description above will suffice for students to get an intuitive idea of what a graph is, which can they later refine.