Graph Theory

  1. Representing Graphs
    1. Adjacency Matrix
      1. Construction and Notation
        1. Matrix Elements
          1. Properties and Interpretation
            1. Symmetry for Undirected Graphs
              1. Sparsity and Density
                1. Diagonal Elements
                2. Operations on Adjacency Matrices
                  1. Matrix Multiplication
                    1. Path Counting
                      1. Powers of Adjacency Matrix
                      2. Space Complexity
                        1. O(n²) Storage
                          1. Memory Considerations
                        2. Adjacency List
                          1. Construction and Notation
                            1. List Structure
                              1. Advantages for Sparse Graphs
                                1. Space Efficiency
                                  1. Neighbor Iteration
                                    1. Dynamic Updates
                                    2. Implementation Considerations
                                    3. Incidence Matrix
                                      1. Construction and Notation
                                        1. Vertex-Edge Relationships
                                          1. Properties and Use Cases
                                            1. Directed vs Undirected Graphs
                                            2. Edge List
                                              1. Construction and Notation
                                                1. Simple Representation
                                                  1. Applications and Limitations
                                                    1. Sorting Considerations
                                                    2. Comparison of Representations
                                                      1. Space Complexity Analysis
                                                        1. Time Complexity for Common Operations
                                                          1. Edge Existence Queries
                                                            1. Neighbor Enumeration
                                                              1. Edge Addition and Deletion
                                                              2. Suitability for Different Graph Types
                                                                1. Dense vs Sparse Graphs
                                                                  1. Static vs Dynamic Graphs

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                                                              2. Fundamental Concepts and Types of Graphs

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                                                              4. Paths, Walks, and Cycles