Summary

This project asks students to define, motivate, and explore an objective measure of the complexity of a graph.

Notes for the instructor

This is an open-ended, capstone-like project designed to come at the end of the graph theory portion of a discrete mathematics course. I like this project because it provides a natural opportunity to discuss the general nature of mathematical research. Based on student feedback, I believe it is particularly successful because it gives students a genuine (and often unexpected) opportunity to express themselves in a mathematics course. Moreover, their insights have led, in some cases, to intriguing, worthwhile, and enjoyable independent student research projects.

At least two one-hour class periods should be set aside for this project. The first hour can be devoted to allowing students to work individually or in groups to devise and experiment with a measure of the complexity of a graph. The second hour can then be devoted to the presentation and discussion of the results. If you have the time and interest, this project may easily be turned into a more substantial research project (which I have done in a graph theory course) by asking the students, at each stage, to explore more fully their measure of complexity.

To get the students going, I think it is worth mentioning that mathematicians and computer scientists have defined the complexity of a graph in several different ways (and for several different purposes).