Start Doing Graph Theory © 2025 by Mikhail Lavrov is licensed under CC BY-SA 4.0
Start Doing Graph Theory is an open-access introduction to graph theory: the mathematical study of objects with relationships between them. If this description sounds like it could describe just about anything, you are right! Graph theory finds uses in other areas of pure math, in applied math and computer science, but also in everyday questions, games, and puzzles. This is why I think a bit of graph theory is useful for anyone to know.
I have taught graph theory classes to different audiences in different formats, but I have based this book mainly on lecture notes that I have used to teach graph theory (MATH 3332) at Kennesaw State University. I have expanded those lecture notes with additional content, detailed explanations of the topics my students struggled with the most, and a healthy dose of reorganization. The result is intended to be suitable both for an undergraduate course on graph theory and for independent self-study.
In case you missed the warning on the main page, I'll include it again: the book is only 50% finished at the moment. I expect it to be finished by the end of October, so my cautious promise is that it will be complete by the end of 2025.
There's a reason I called this book Start Doing Graph Theory: you can't learn mathematics without doing it. I can't, actually, make you do anything, but I have tried to write the book to make it easier for you to do math. I hope to lead by example in this, and show you how the graph theory you're learning can be used to solve problems someone might actually care about, whenever possible. Throughout the text, I have included the kind of questions that I might ask if I were teaching you graph theory; even though I'm not there, I recommend that you stop and think before reading the answer.
This book is not just an introduction to graph theory; it is a proof-based introduction to graph theory. Many people learning graph theory for the first time are new to writing proofs, and I have written the book with this in mind. In addition to the two chapters specifically about proof-writing, I try to say a little bit in each proof about why we do what we do, how we could have thought of it, and what we must be careful of.
The pronoun "we" is common in mathematical writing, and I think this makes sense. When we prove a theorem, we do it together, and it should be an active process on your part. You should keep our common goals in mind, and think about how to get from where we are to where we'd like to be. If I present an argument, your job is to be as skeptical as possible: we have only proven a theorem once you are convinced that the proof I wrote down is airtight, or once you've done your own work to fill in any gaps.
Both the book Start Doing Graph Theory and this website are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. In particular, the book is free to download and read, and will remain free to download and read. I will let the Creative Commons website describe the details more precisely.
If so, please get in touch! I would like to hear from all of my readers, so please write to me at [email protected] whether you've used it as a course textbook, or merely stumbled on it randomly and decided to read it, or anything in between. If you're thinking, "But surely he doesn't mean me," then you're wrong; I do mean you. If you're thinking, "I bet he gets too much email already," then don't worry about it; if too many people write to me, I'll change this paragraph accordingly.
Start Doing Graph Theory © 2025 by Mikhail Lavrov is licensed under CC BY-SA 4.0