Graph Theory

Graph Theory is a branch of mathematics that studies the properties and structures of graphs, which are mathematical representations of a set of objects connected by edges. In a graph, the objects are represented as vertices (or nodes), and the connections between them are represented as edges (or lines). This area of study encompasses various concepts such as connectivity, paths, cycles, and traversability, and finds applications in diverse fields including computer science, biology, social sciences, and transportation networks. Graph Theory provides essential tools for analyzing relationships and solving problems related to networks, optimization, and combinatorics.

1. Basic Concepts in Graph Theory
   1. Graphs
      1. Undirected Graphs
         1. Definition
         2. Properties
         3. Use Cases
      2. Directed Graphs (Digraphs)
         1. Definition
         2. Properties
         3. Use Cases
      3. Weighted Graphs
         1. Definition
         2. Edge Weights
            1. Positive Weights
            2. Negative Weights
         3. Applications
      4. Simple Graphs
         1. Definition
         2. Characteristics
            1. Without Loops
            2. Without Multiple Edges
         3. Examples
      5. Multigraphs
         1. Definition
         2. Characteristics
            1. Multiple Edges Allowed
         3. Use Cases
      6. Pseudographs
         1. Definition
         2. Characteristics
            1. Loops Allowed
            2. Multiple Edges Allowed
         3. Use Cases
   2. Components of Graphs
      1. Vertices (Nodes)
         1. Definition
         2. Properties
         3. Role in Graph Structure
      2. Edges (Links)
         1. Definition
         2. Types of Edges
            1. Directed Edge
            2. Undirected Edge
         3. Representation
      3. Degree of a Vertex
         1. Definition
         2. Types
            1. In-degree (for Digraphs)
            2. Out-degree (for Digraphs)
         3. Average Degree
         4. Graphically Representing Degree Distribution
      4. Adjacent Vertices
         1. Definition
         2. Neighborhood of a Vertex
         3. Applications in Algorithms
      5. Incident Edges
         1. Definition
         2. Relationship with Degree
         3. Visual Identification in Graphs
   3. Subgraphs
      1. Definition
      2. Induced Subgraphs
      3. Spanning Subgraphs
      4. Application Examples
   4. Paths and Cycles
      1. Paths
         1. Definition
         2. Simple Paths
         3. Non-Simple Paths
         4. Path Length
      2. Cycles
         1. Definition
         2. Simple Cycles
         3. Directed and Undirected Cycles
      3. Hamiltonian Paths and Cycles
         1. Definition
         2. Characteristics
         3. Problems and Algorithms
      4. Eulerian Paths and Cycles
         1. Definition
         2. Characteristics
         3. Eulerian Circuits
         4. Necessary and Sufficient Conditions
   5. Connectivity
      1. Connected Graphs
         1. Definition
         2. Strongly Connected Components (SCC)
      2. Strongly Connected Graphs
         1. Definition
         2. Applications
      3. Weakly Connected Graphs
         1. Definition
         2. Difference from Strongly Connected Graphs
      4. Biconnected Graphs
         1. Biconnected Components
         2. Articulation Points
         3. Applications in Network Design
   6. Graph Representations
      1. Adjacency Matrix
         1. Definition
         2. Characteristics
         3. Memory Consumption
         4. Advantages and Disadvantages
      2. Adjacency List
         1. Definition
         2. Properties
         3. Memory Efficiency
         4. Advantages and Disadvantages
      3. Incidence Matrix
         1. Definition
         2. Properties
         3. Comparison with Other Representations