5 The Mathematics of Getting Around CONCLUSION

Themes of Chapter 5 In this chapter we got our first introduction to three fundamental ideas. First, we learned about a simple but powerful concept for describing relationships within a set of objects – the concept of a graph. This idea can be traced back to Euler, some 270 years ago. Since then, the study of graphs has grown into one of the most important and useful branches of modern mathematics.

Themes of Chapter 5 The second important idea of this chapter is the concept of a graph model. Every time we take a real-life problem and turn it into a mathematical problem, we are, in effect, modeling. In this chapter we learned about a new type of modeling called graph modeling, in which we use graphs and the mathematical theory of graphs to solve certain types of routing problems.

Themes of Chapter 5 By necessity, the routing problems that we solved in this chapter were fairly simplistic – crossing a few bridges, patrolling a small neighborhood, designing a parade route – what’s all the fuss about? We should not be deceived by the simplicity of these examples – larger-scale variations on these themes have significant practical importance.

Themes of Chapter 5 In many big cities, where the efficient routing of municipal services (police patrols, garbage collection, etc.) is a major issue, the very theory that we developed in this chapter is being used on a large scale, the only difference being that many of the more tedious details are mechanized and carried out by a computer.

Themes of Chapter 5 The third important concept introduced in this chapter is that of an algorithm – a set of procedural rules that, when followed, provide solutions to certain types of problems. In this chapter we learned about Fleury’s algorithm, which helps us find an Euler circuit or an Euler path in a graph.

Themes of Chapter 5 When it comes to algorithms of any kind, be they for doing arithmetic calculations or for finding circuits in graphs, there is one standard piece of advice that always applies: Practice makes perfect.