Graph Theory

1. Advanced Topics in Graph Theory
   1. Spectral Graph Theory
      1. Graph Laplacian
         1. Definition and Properties
         2. Laplacian Matrix
            1. Symmetric
            2. Diagonal Matrix Subtraction
         3. Applications
            1. Network Analysis
            2. Image Segmentation
         4. Laplacian Eigenmaps
            1. Dimensionality Reduction
            2. Data Clustering
      2. Eigenvalues and Eigenvectors
         1. Characteristic Polynomial
         2. Algebraic Connectivity
         3. Spectral Gap
         4. Spectral Radius
         5. Applications
            1. Stability Analysis of Networks
            2. Graphical Clustering
   2. Algebraic Graph Theory
      1. Adjacency Polynomials
         1. Generating Functions
         2. Roots and Coefficients
         3. Applications to Graph Recognition
      2. Chemin Polynomial
         1. Definition and Properties
         2. Chemical Applications
         3. Computational Aspects
      3. Graph Automorphisms
         1. Orbit Structure
         2. Symmetry Properties
   3. Topological Graph Theory
      1. Graph Embeddings
         1. Embedding Graphs in Surfaces
            1. Planar Embeddings
            2. Embeddings on Higher-Genus Surfaces
         2. Knot Theory
      2. Graph Minors
         1. Minor-Closed Properties
         2. Wagner’s Theorems
         3. Applications
            1. Network Design
            2. Computational Complexity
   4. Random Graphs
      1. Erdős–Rényi Model
         1. Probability Models
         2. Graph Properties
            1. Connectivity
            2. Giant Component
         3. Phase Transitions
      2. Barabási–Albert Model
         1. Scale-Free Networks
         2. Preferential Attachment
         3. Power-Law Degree Distribution
      3. Other Models
         1. Stochastic Block Models
         2. Configuration Models
   5. Extremal Graph Theory
      1. Turán's Theorem
      2. Erdős–Stone Theorem
      3. Applications
         1. Combinatorial Optimization
         2. Network Safety
      4. Hypergraph Extremality
         1. Generalizations to Hypergraphs
   6. Graph Homomorphisms
      1. Homomorphic Properties
         1. Surjective, Injective, Bijective
      2. Applications
         1. Coloring Problems
         2. Network Alignment
      3. Homomorphism Densities
         1. Theory and Calculations
   7. Ramsey Theory in Graphs
      1. Ramsey Numbers
         1. Definition and Properties
         2. Known Results and Bounds
      2. Applications
         1. Combinatorial Design Problems
      3. Generalizations
         1. Hypergraph Ramsey Theory
   8. Graphs in Computational Complexity
      1. P vs NP Problem
         1. Definitions and Implications
         2. Importance in Graph Theory
      2. NP-complete Graph Problems
         1. Graph Coloring
         2. Hamiltonian Path Problem
         3. Clique Problem