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Graph Theory

G(V,E)

Graph Theory

Vertices, edges, paths, and network algorithms

198Modules

10Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Discrete Mathematics is the natural lead-in — graph theory builds on its logic and proof techniques.

Discrete Mathematics

What is Graph Theory?INTRO

Graphs Are EverywhereINTRO

Vertices and EdgesLESSON

Counting Edges ExamplesEXAMPLE

Graph NotationLESSON

Graph Builder ToolACTIVITY

Graph Basics PracticePRACTICE

The Seven Bridges of KönigsbergINTRO

Degree of a VertexLESSON

Degree Counting PracticePRACTICE

The Handshaking LemmaLESSON

Special GraphsLESSON

Complete GraphsLESSON

Social Networks as GraphsREAL WORLD

Bipartite GraphsLESSON

SubgraphsLESSON

Graph IsomorphismLESSON

Isomorphism Testing ExamplesEXAMPLE

Isomorphism CheckerACTIVITY

Introduction to Graphs SummarySUMMARY

Graph Basics ReferenceREFERENCE

Introduction to Graphs QuizQUIZ

Walking Through GraphsINTRO

Walks, Trails, and PathsLESSON

Finding Paths ExamplesEXAMPLE

Paths PracticePRACTICE

Connected vs DisconnectedINTRO

Connected GraphsLESSON

Connected ComponentsLESSON

Cut Vertices and BridgesLESSON

Path Finding ToolACTIVITY

Distance and DiameterLESSON

Shortest PathsLESSON

Shortest Paths PracticePRACTICE

Dijkstra's AlgorithmLESSON

Dijkstra's Algorithm TraceEXAMPLE

Dijkstra SimulatorACTIVITY

Bellman-Ford AlgorithmLESSON

Internet RoutingREAL WORLD

GPS NavigationREAL WORLD

Paths and Connectivity SummarySUMMARY

Paths ReferenceREFERENCE

Paths and Connectivity QuizQUIZ

What are Trees?INTRO

Tree Definition and PropertiesLESSON

Tree BuilderACTIVITY

Characterizations of TreesLESSON

Trees PracticePRACTICE

Cayley's FormulaLESSON

Counting Trees ExamplesEXAMPLE

Trees in Nature and ComputingINTRO

Rooted TreesLESSON

Binary TreesLESSON

Phylogenetic TreesREAL WORLD

Spanning TreesLESSON

Minimum Spanning TreesLESSON

Network DesignREAL WORLD

Prim's AlgorithmLESSON

Kruskal's AlgorithmLESSON

MST Construction ExamplesEXAMPLE

MST VisualizerACTIVITY

MST PracticePRACTICE

Trees SummarySUMMARY

Trees ReferenceREFERENCE

The Eulerian ProblemINTRO

Eulerian TrailsLESSON

Eulerian Graph ExamplesEXAMPLE

Eulerian CircuitsLESSON

Euler's TheoremLESSON

Eulerian PracticePRACTICE

Hierholzer's AlgorithmLESSON

Euler Path FinderACTIVITY

The Hamiltonian ProblemINTRO

Hamiltonian PathsLESSON

Hamiltonian Graph ExamplesEXAMPLE

Hamiltonian CyclesLESSON

Necessary ConditionsLESSON

Hamiltonian PracticePRACTICE

Ore and Dirac TheoremsLESSON

Hamilton Path ExplorerACTIVITY

The Traveling SalesmanINTRO

Traveling Salesman ProblemLESSON

Route Planning ApplicationsREAL WORLD

Eulerian and Hamiltonian SummarySUMMARY

Eulerian/Hamiltonian ReferenceREFERENCE

Eulerian and Hamiltonian QuizQUIZ

The Coloring ProblemINTRO

Vertex ColoringLESSON

Graph Coloring ToolACTIVITY

Vertex Coloring PracticePRACTICE

Chromatic NumberLESSON

Finding Chromatic NumbersEXAMPLE

Chromatic Number CalculatorACTIVITY

Greedy Coloring AlgorithmLESSON

Brooks' TheoremLESSON

Register AllocationREAL WORLD

Chromatic PolynomialLESSON

Chromatic Polynomial ExamplesEXAMPLE

Chromatic Polynomial PracticePRACTICE

Edge ColoringLESSON

Vizing's TheoremLESSON

Scheduling ProblemsREAL WORLD

Map Coloring HistoryINTRO

The Four Color TheoremLESSON

Coloring Planar GraphsLESSON

Graph Coloring SummarySUMMARY

Graph Coloring ReferenceREFERENCE

Graph Coloring QuizQUIZ

What are Planar Graphs?INTRO

Planar Graph DefinitionLESSON

Planar EmbeddingsINTRO

Planar Embedding ToolACTIVITY

Euler's FormulaLESSON

Euler Formula ExamplesEXAMPLE

Euler Formula PracticePRACTICE

Consequences of Euler's FormulaLESSON

K₅ and K₃,₃ are Non-PlanarLESSON

Platonic SolidsLESSON

Circuit Board DesignREAL WORLD

Kuratowski's TheoremLESSON

Planarity Testing AlgorithmsLESSON

Planarity Testing ExamplesEXAMPLE

Planarity TesterACTIVITY

Planarity PracticePRACTICE

Dual GraphsLESSON

Planar Graphs SummarySUMMARY

Planar Graphs ReferenceREFERENCE

Planar Graphs QuizQUIZ

The Matching ProblemINTRO

Matchings DefinitionLESSON

Matching Problem ExamplesEXAMPLE

Maximum MatchingsLESSON

Matching SolverACTIVITY

Bipartite MatchingLESSON

Hall's Marriage TheoremLESSON

Matchings PracticePRACTICE

Hungarian AlgorithmLESSON

Job AssignmentREAL WORLD

Flow in NetworksINTRO

Network Flow BasicsLESSON

Network Flow ExamplesEXAMPLE

Ford-Fulkerson AlgorithmLESSON

Network Flow SimulatorACTIVITY

Max-Flow Min-Cut TheoremLESSON

Network Flow PracticePRACTICE

Applications of Network FlowLESSON

Transportation NetworksREAL WORLD

Matchings and Flow SummarySUMMARY

Matchings and Flow ReferenceREFERENCE

Matchings and Flow QuizQUIZ

Introduction to Directed GraphsINTRO

Digraph BasicsLESSON

Digraph BuilderACTIVITY

In-Degree and Out-DegreeLESSON

Digraphs PracticePRACTICE

Directed Paths and CyclesLESSON

Strong ConnectivityLESSON

Finding Strongly Connected ComponentsLESSON

SCC Finding ExamplesEXAMPLE

TournamentsLESSON

Where Digraphs AppearINTRO

Directed Acyclic GraphsLESSON

Topological SortingLESSON

Topological Sort ExamplesEXAMPLE

Topological Sort VisualizerACTIVITY

Topological Sort PracticePRACTICE

Dependency ManagementREAL WORLD

Web Crawling and PageRankREAL WORLD

Directed Graphs SummarySUMMARY

Directed Graphs ReferenceREFERENCE

Directed Graphs QuizQUIZ

Algebra Meets Graph TheoryINTRO

Matrices from GraphsINTRO

The Adjacency MatrixLESSON

Matrix Computation ExamplesEXAMPLE

Graph Matrix CalculatorACTIVITY

Properties of Adjacency MatrixLESSON

The Incidence MatrixLESSON

Graph Matrices PracticePRACTICE

The Laplacian MatrixLESSON

Eigenvalues of GraphsLESSON

Eigenvalue Analysis ExamplesEXAMPLE

Spectral PropertiesLESSON

Spectral AnalyzerACTIVITY

Spectral Methods PracticePRACTICE

Algebraic ConnectivityLESSON

Spectral ClusteringLESSON

Google PageRankREAL WORLD

Algebraic Graph Theory SummarySUMMARY

Algebraic Graph Theory ReferenceREFERENCE

Algebraic Graph Theory QuizQUIZ

Problem Solving GuideSUMMARY

Graph Theory Master ReferenceREFERENCE

Complete Formula SheetFORMULA SHEET

Graph Algorithms GuideFORMULA SHEET

Comprehensive Graph Theory TestQUIZ