Eulerian Paths and the Bridges of Königsberg
Explore how Leonhard Euler's solution to the famous Königsberg bridge problem laid the foundations for graph theory and topology.
Dijkstra's Algorithm: Shortest Paths in Weighted Graphs
A comprehensive breakdown of Dijkstra's algorithm, its greedy approach, time complexity analysis, and real-world routing applications.
Graph Coloring and the Four Color Theorem
Understanding chromatic numbers, Brooks' theorem, and the computer-assisted proof that revolutionized mathematical verification.
Social Network Analysis: Centrality Measures
Degree, betweenness, closeness, and eigenvector centrality explained with practical examples from social media and epidemiology.
Planar Graphs and Topological Embeddings
Study graphs that can be drawn without edge crossings, Euler's formula for planar graphs, and Kuratowski's characterization theorem.
Graph Neural Networks in Machine Learning
How message-passing architectures and spectral methods transform non-Euclidean data into learnable representations for AI.
Bipartite Matching and the Hungarian Algorithm
Fundamentals of two-sided matching problems, augmenting paths, and how the Hungarian method solves assignment problems optimally.
Random Graphs and Erdős–Rényi Models
Probability thresholds, phase transitions, and the emergence of giant components in G(n,p) and G(n,m) random graph models.