Graph Theory

  1. Trees and Forests
    1. Definitions and Properties
      1. Tree Definition
        1. Acyclic Connected Graph
          1. Minimally Connected Graph
            1. Maximally Acyclic Graph
            2. Equivalent Characterizations
              1. Unique Path Between Any Two Vertices
                1. Connected with n-1 Edges for n Vertices
                  1. Acyclic with n-1 Edges for n Vertices
                    1. Adding Any Edge Creates Exactly One Cycle
                    2. Forest Definition
                      1. Collection of Disjoint Trees
                        1. Acyclic Graph
                      2. Rooted and Unrooted Trees
                        1. Rooted Trees
                          1. Root Vertex Selection
                            1. Parent-Child Relationships
                              1. Ancestor and Descendant Relations
                                1. Sibling Relationships
                                  1. Leaves and Internal Nodes
                                    1. Tree Height and Depth
                                      1. Subtrees
                                      2. Unrooted Trees
                                        1. No Distinguished Root
                                          1. Symmetric Structure
                                            1. Conversion to Rooted Trees
                                          2. Spanning Trees
                                            1. Definition
                                              1. Subgraph Properties
                                                1. Connectivity Preservation
                                                2. Existence in Connected Graphs
                                                  1. Number of Spanning Trees
                                                    1. Cayley's Formula for Complete Graphs
                                                      1. Counting in General Graphs
                                                      2. Matrix Tree Theorem
                                                        1. Laplacian Matrix
                                                          1. Cofactor Calculation
                                                        2. Minimum Spanning Trees (MST)
                                                          1. Problem Definition
                                                            1. Weight Minimization
                                                              1. Connectivity Preservation
                                                              2. Properties of MSTs
                                                                1. Cut Property
                                                                  1. Cycle Property
                                                                    1. Uniqueness Conditions
                                                                    2. Algorithms
                                                                      1. Kruskal's Algorithm
                                                                        1. Edge Sorting Approach
                                                                          1. Union-Find Data Structure
                                                                            1. Complexity Analysis
                                                                            2. Prim's Algorithm
                                                                              1. Vertex Growing Approach
                                                                                1. Priority Queue Implementation
                                                                                  1. Complexity Analysis
                                                                                  2. Borůvka's Algorithm
                                                                                    1. Component-Based Approach
                                                                                      1. Parallel Implementation
                                                                                    2. Applications
                                                                                      1. Network Design
                                                                                        1. Clustering
                                                                                          1. Approximation Algorithms

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                                                                                      5. Graph Traversal

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                                                                                      7. Shortest Path Algorithms