Free exploration sandbox and test your knowledge
Welcome to the final module! You've journeyed through graph foundations, trees and properties, paths and coloring, network flow, spectral theory, and random networks. Now it's time to consolidate your knowledge through hands-on experimentation and comprehensive testing.
Use the interactive playground to build graphs and run algorithms. Then take the comprehensive quiz to test your understanding across all topics. Are you ready to become a graph theory expert?
Build your own graphs and explore algorithms! Click to add vertices, connect them with edges, and run various graph algorithms to see them in action.
Test your knowledge! This comprehensive quiz covers all topics from foundations to advanced network theory. Choose your difficulty level and see how well you understand graphs.
Use induction on the number of edges to prove V - E + F = 2 for connected planar graphs. What happens when you remove an edge?
Prove that the complete graph K₅ cannot be drawn in the plane without edge crossings. Use Euler's formula and the edge-face inequality.
Find the chromatic polynomial P(G, k) for a cycle C₄. How many ways can you color it with k colors? Verify that P(C₄, 2) = 2 (only 2 proper 2-colorings exist).
Construct a graph that is Eulerian but not Hamiltonian. Then construct one that is Hamiltonian but not Eulerian. Can a graph be both?
For the complete graph Kₙ, find all eigenvalues of the Laplacian matrix. Why is the spectral gap λ₂ = n maximal among all graphs on n vertices?
Determining whether two graphs are structurally identical. Famously solved in quasi-polynomial time by Babai (2015) but not known to be in P.
Machine learning on graph-structured data. Used in drug discovery, social network analysis, and recommendation systems.
Sparse graphs with strong connectivity properties. Applications in error-correcting codes, cryptography, and derandomization.
Using group theory to study graph symmetries. Cayley graphs connect group structure to graph properties.
You've completed the Graph Theory learning module! You now understand fundamental concepts, can analyze graph properties, implement algorithms, and have the foundation to explore advanced topics.
Mastered: Vertices, edges, degree, connectivity, trees, spanning trees
Explored: BFS, DFS, Dijkstra, Eulerian/Hamiltonian paths
Applied: Graph coloring, planarity testing, network flow
Advanced: Spectral theory, random graphs, scale-free networks
Practiced: Interactive building, algorithm visualization, comprehensive quiz