At this point I hope you've gotten some appreciation for the subject and its applications as well as an idea of how mathematicians specialize. At no point did we use a geometric idea more complicated than a segment but we still came up with thirty-one pages worth of material and what we saw was by no means the whole story. For example, we talked almost exclusively about graphs. How does the concept of a trail change if we modify the definition to use digraphs instead?
If you want to learn more I highly recommend Gary Chartrand's Introduction to Graph Theory. The book is theoretical in the sense that it includes theorems and their proofs but it follows a pattern similar to this classroom's - the mathematical ideas are presented side-by-side with real world applications. Mr. Chartrand's writing style is very down-to-earth so a high school students should find his work challenging but understandable.
If you enjoyed this class and its format, please continue to check the White Crane Education home page. We're currently working hard on our algebra curriculum and will definitely have free resources similar to this one. In our spare time, we're also working on our next geometry classroom. They'll all be announced on the home page as soon as they're available.