Familiarity with Graph Theory
This syllabus covers a wide range of topics in graph theory, from the basic definitions and terminology to advanced concepts and applications. It includes modules on graph algorithms, combinatorial aspects of graph theory, algebraic and structural graph theory, random graphs and probabilistic methods, and various applications of graph theory in different fields.
By following this syllabus, learners will gain a solid foundation in graph theory and develop the necessary skills to apply graph-theoretic concepts and techniques to solve real-world problems. The modular structure allows for flexibility in learning, enabling learners to focus on specific areas of interest while still maintaining a comprehensive understanding of the subject.
Introduction to Graph Theory (20 modules):
1-4: Definition of graphs, vertices, edges, and basic terminology
5-8: Types of graphs (directed, undirected, weighted, bipartite, etc.)
9-12: Graph representations (adjacency matrix, adjacency list, edge list)
13-16: Subgraphs, induced subgraphs, and graph isomorphism
17-20: Paths, cycles, and connectivity in graphs
Graph Traversal or graph search and NP-hard algorithms such as the Traveling Salesman Problem, maximum flow problems maze generation problem (30 modules):
21-24: Depth-First Search (DFS) and its applications
25-28: Breadth-First Search (BFS) and its applications
29-32: Shortest path algorithms (Dijkstra, Bellman-Ford, Floyd-Warshall)
33-36: Minimum Spanning Trees (Kruskal’s and Prim’s algorithms)
37-40: Topological sorting and its applications
41-44: Strongly Connected Components (Kosaraju’s and Tarjan’s algorithms)
45-50: Network flow problems (Ford-Fulkerson, Max Flow-Min Cut Theorem)
Graph Theory and Combinatorics (30 modules):
Graphs are fundamental objects in combinatorics. Considerations of graph theory range from enumeration (e.g., the number of graphs on n vertices with k edges) to existing structures (e.g., Hamiltonian cycles) to algebraic representations (e.g., given a graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation? The combinatorial interpretation of the Tutte polynomial provides a powerful tool for studying the structure and properties of undirected graphs, and it has applications in various areas, including network reliability, quantum computation and statistical physics, and knot theory and knot mathematics.). Although there are very strong connections between graph theory and combinatorics, they are sometimes thought of as separate subjects. While combinatorial methods apply to many graph theory problems, the two disciplines are generally used to seek solutions to different types of problems.