Graph Theory· Study notes
Discrete and Data
Networks from the ground up: degrees and isomorphism, connectivity, trees and spanning trees, shortest paths and minimum spanning trees, Euler and Hamilton tours, matchings, planarity, coloring, network flows, and Ramsey theory.
What you'll be able to do
- Model problems as graphs and prove facts with degree counting
- Run and justify BFS, Dijkstra, Kruskal and Prim
- Decide Eulerian, Hamiltonian, bipartite and planar questions
- Use Hall, max-flow min-cut, coloring bounds and Ramsey arguments
Read the notes as a guided lesson
Walk the notes one box at a time — prose, worked examples, quick checks — with a timed practice round after every section. For the full course with interactive lessons, go back to the course.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics
Isomorphism and Subgraphs
6 lessonsWhen two graphs are the same, and the invariants that prove they are not
Walks, Paths and Connectivity
6 lessonsWalks and paths, components, counting walks with matrices, and weak points of a network
Trees
6 lessonsThe many equivalent definitions of a tree, spanning trees, and counting labeled trees
Graph Algorithms
5 lessonsSearching a graph, shortest paths, and minimum spanning trees
Eulerian and Hamiltonian Graphs
6 lessonsTraversing every edge, visiting every vertex, and the theorems that decide when you can
Bipartite Graphs and Matchings
6 lessonsTwo-sided graphs, matchings, Hall's marriage theorem, and stable matchings
Planar Graphs
6 lessonsDrawing without crossings, Euler's formula, and the two forbidden graphs
Graph Coloring
6 lessonsChromatic number, greedy coloring, the chromatic polynomial, and edge coloring
Directed Graphs and Network Flows
6 lessonsDigraphs and tournaments, strong connectivity, and the max-flow min-cut theorem
Extremal Graph Theory and Ramsey Theory
6 lessonsHow many edges force a triangle, and why complete disorder is impossible