A complete course
Graph Theory: From Zero to Diestel
A self-contained course that takes you from "what is a graph?" to the frontier of the subject, and then into how graphs power modern artificial intelligence. Built on Reinhard Diestel's Graph Theory (6th edition, Springer, 2024), the standard graduate text, but assuming no mathematical notation at all: every symbol is introduced, read aloud, and illustrated before it is used.
How this course works
Diestel's book is written for mathematics graduate students. It is precise, beautiful, and, for most readers, unreadable: it assumes you already speak the language of sets, functions, and proofs. This course removes that barrier. Each chapter here mirrors a chapter of the book and does four things:
1. Teaches the notation first. Every chapter opens with a table of the new symbols it uses, each with a "read it as" pronunciation and a tiny example. A master table collecting all of them is in the notation reference.
2. Motivates before defining. Every definition is preceded by the question it answers. Every theorem comes with the intuition for why it is true, in plain language, before any symbols.
3. Illustrates everything. Each concept gets a picture. Each theorem gets a worked example on a small concrete graph, plus a real-world application: kidney exchanges, compiler design, epidemic models, chip layout.
4. Tests you. Each chapter ends with a multiple-choice quiz with instant feedback and explanations.
Work through the chapters in order. Later chapters use earlier ones. Chapters 0 through 6 are the foundation everyone should have; chapters 7 through 12 are the deep theory; chapters 13 through 15 leave the book behind and show graph theory at work inside today's AI systems, with citations to current research.
How to Read Mathematics
Sets, functions, and the language of proofs. The notation bootstrap: everything Diestel assumes on page one, taught from scratch.
Chapter 1The Basics
What a graph is. Vertices, edges, degree, paths, cycles, trees, bipartite graphs, minors, Euler tours, and the linear algebra of graphs.
Matching, Covering and Packing
Pairing things up optimally: marriage theorems, kidney exchanges, and the combinatorics of fair assignment.
Chapter 3Connectivity
How robust is a network? Menger's theorem and the exact price of disconnecting a graph.
Chapter 4Planar Graphs
Which graphs can be drawn without crossings? Euler's formula, Kuratowski's theorem, and why circuit boards care.
Chapter 5Colouring
The four colour problem, scheduling conflicts, compiler registers, and the chromatic number.
Chapter 6Flows
Max-flow, min-cut: how much can move through a network, and the surprising algebra of circulations.
Extremal Graph Theory
How many edges force a structure to appear? Turán's theorem and Szemerédi's regularity lemma.
Chapter 8Infinite Graphs
What survives when graphs become infinite: compactness, ends, and infinity done carefully.
Chapter 9Ramsey Theory
Complete disorder is impossible: patterns that no amount of scrambling can destroy.
Chapter 10Hamilton Cycles
Tours that visit every vertex exactly once, and the conditions that guarantee them.
Chapter 11Random Graphs
Graphs by coin flip: the probabilistic method, thresholds, and why random networks have sharp phase changes.
Chapter 12Graph Minors, Tree-Width and Tangles
The deepest theorem in the subject: well-quasi-ordering, tree decompositions, and where hard problems become easy.
Graphs in Artificial Intelligence
Graph neural networks, knowledge graphs, transformers as graphs, and why the adjacency matrix runs the modern world.
Chapter 14GraphRAG: Retrieval over Knowledge Graphs
Why vector search alone fails at global questions, and how community detection over entity graphs fixes it.
Chapter 15Trustworthy AI, One Graph at a Time
Grounding, provenance, causal graphs, and attribution circuits: using graph structure to make AI systems verifiable.
Master Notation Table
Every symbol in the course, in one sortable place, with chapter links.
Primary source
- R. Diestel, Graph Theory, 6th edition, Graduate Texts in Mathematics 173, Springer (2024). doi:10.1007/978-3-662-70107-2. The author's book site is diestel-graph-theory.com.
Chapters 13–15 cite their own research literature inline. This course is an independent teaching companion; all theorem attributions follow the book.