Graph Theory

  1. Advanced Topics in Graph Theory
    1. Spectral Graph Theory
      1. Matrix Representations
        1. Adjacency Matrix Eigenvalues
          1. Spectral Properties
            1. Characteristic Polynomial
            2. Laplacian Matrix
              1. Definition and Construction
                1. Properties
                  1. Positive Semidefinite
                    1. Kernel and Rank
                    2. Normalized Laplacian
                    3. Spectral Graph Properties
                      1. Connectivity and Eigenvalues
                        1. Cheeger's Inequality
                          1. Isoperimetric Problems
                            1. Expansion and Conductance
                            2. Spectral Clustering
                              1. Random Walks on Graphs
                              2. Applications
                                1. Graph Partitioning
                                  1. Community Detection
                                    1. Dimensionality Reduction
                                  2. Extremal Graph Theory
                                    1. Turán-Type Problems
                                      1. Turán's Theorem
                                        1. Statement and Proof
                                          1. Turán Graphs
                                            1. Extremal Functions
                                            2. Forbidden Subgraph Problems
                                              1. Zarankiewicz Problem
                                              2. Ramsey Theory
                                                1. Ramsey Numbers
                                                  1. Classical Ramsey Numbers R(s,t)
                                                    1. Known Values and Bounds
                                                    2. Ramsey's Theorem
                                                      1. Finite and Infinite Versions
                                                        1. Proof Techniques
                                                        2. Applications
                                                          1. Combinatorial Geometry
                                                            1. Number Theory
                                                        3. Random Graphs
                                                          1. Erdős-Rényi Model
                                                            1. G(n,p) Model Definition
                                                              1. G(n,m) Model Definition
                                                                1. Relationship Between Models
                                                                2. Threshold Functions
                                                                  1. Definition and Significance
                                                                    1. Sharp Thresholds
                                                                      1. Coarse Thresholds
                                                                      2. Properties of Random Graphs
                                                                        1. Connectivity Threshold
                                                                          1. Giant Component Emergence
                                                                            1. Chromatic Number
                                                                              1. Clique Number
                                                                              2. Evolution of Random Graphs
                                                                                1. Phase Transitions
                                                                                  1. Critical Phenomena
                                                                                    1. Scaling Laws
                                                                                  2. Graph Decompositions
                                                                                    1. Edge Decompositions
                                                                                      1. Path Decompositions
                                                                                        1. Cycle Decompositions
                                                                                          1. Star Decompositions
                                                                                          2. Tree Decompositions
                                                                                            1. Definition and Properties
                                                                                              1. Tree Width
                                                                                                1. Computational Significance
                                                                                                2. Path Width
                                                                                                  1. Applications in Algorithms
                                                                                                  2. Other Decompositions
                                                                                                    1. Modular Decomposition
                                                                                                      1. Clique Decomposition
                                                                                                        1. Block Decomposition
                                                                                                      2. Algebraic Graph Theory
                                                                                                        1. Group Actions on Graphs
                                                                                                          1. Automorphism Groups
                                                                                                            1. Orbit-Stabilizer Theorem
                                                                                                              1. Vertex-Transitive Graphs
                                                                                                                1. Edge-Transitive Graphs
                                                                                                              2. Graph Homomorphisms
                                                                                                                1. Definition and Properties
                                                                                                                  1. Homomorphism Order
                                                                                                                    1. Core Graphs
                                                                                                                      1. Applications
                                                                                                                        1. Graph Coloring
                                                                                                                          1. Constraint Satisfaction
                                                                                                                        2. Cayley Graphs
                                                                                                                          1. Construction from Groups
                                                                                                                            1. Examples and Applications
                                                                                                                            2. Strongly Regular Graphs
                                                                                                                              1. Definition and Parameters
                                                                                                                                1. Existence Conditions
                                                                                                                                  1. Examples and Constructions

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                                                                                                                              11. Matchings

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                                                                                                                              1. Introduction to Graph Theory