Graph Theory for Computer Science

Graphs model relationships and interactions between entities. They are central to algorithms, systems, networking, AI, and compilers.

Why Graphs Are Everywhere in CS

  • Internet routing and network topology
  • Dependency graphs in package managers and build tools
  • Knowledge graphs and recommendation systems
  • State-transition systems and automata
  • Control-flow and call graphs in compilers

Course Structure

  1. Graph definitions and data structures
  2. Traversal algorithms and invariants
  3. Trees and spanning structures
  4. Shortest path algorithms
  5. Connectivity and flow
  6. System-level applications

Core Algorithm Map

flowchart LR
  A[Graph Representation] --> B[BFS/DFS]
  B --> C[Connected Components]
  B --> D[Topological Sort]
  A --> E[Weighted Graphs]
  E --> F[Dijkstra]
  E --> G[Bellman-Ford]
  E --> H[Floyd-Warshall]
  A --> I[Undirected Weighted]
  I --> J[MST: Kruskal/Prim]
  B --> K[SCC]
  K --> L[Flow and Matching]

Proof Skills for Graph Algorithms

  • Loop invariants (Dijkstra correctness)
  • Exchange arguments (MST correctness)
  • Reachability and induction on BFS layers
  • Cut/cycle properties for weighted graphs

Exercise Set (Warm-up)

  1. Give one real-world system modeled naturally as directed graph.
  2. Give one modeled as undirected weighted graph.
  3. Explain why adjacency list is preferred for sparse graphs.

Summary

Graph theory converts structure into computation. This section focuses on both formal correctness and implementation decisions.