Graph Theory

Graph theory is a branch of mathematics concerned with the study of graphs, which are abstract structures consisting of a set of objects called vertices (or nodes) connected by links called edges. Rather than focusing on data visualization, this field analyzes the properties of networks, such as connectivity, paths, and cycles, to model pairwise relationships between objects. Its principles are fundamental to solving a vast array of problems, from finding the shortest navigation route and designing resilient computer networks to understanding molecular structures and analyzing the spread of information through social networks. By abstracting complex systems into a simplified model of points and lines, graph theory provides a powerful tool for understanding the structure and dynamics of interconnected systems.

1. Introduction to Graph Theory
   1. Core Definitions
      1. Vertex
         1. Definition and Notation
         2. Properties of Vertices
         3. Vertex Sets
      2. Edge
         1. Definition and Notation
         2. Directed Edges
         3. Undirected Edges
         4. Weighted Edges
         5. Unweighted Edges
      3. Graph as Ordered Pair (V, E)
         1. Formal Mathematical Definition
         2. Vertex Set Notation
         3. Edge Set Notation
         4. Examples of Graph Representations
   2. Historical Context
      1. Seven Bridges of Königsberg
         1. Problem Statement
         2. Euler's Solution
         3. Historical Significance
      2. Early Developments
         1. 18th Century Origins
         2. 19th Century Progress
      3. Key Contributors
         1. Leonhard Euler
         2. Gustav Kirchhoff
         3. Arthur Cayley
   3. Applications of Graph Theory
      1. Computer Science
         1. Data Structures
         2. Algorithm Design
         3. Computer Networks
         4. Database Design
         5. Compiler Design
      2. Social Networks
         1. Relationship Modeling
         2. Community Detection
         3. Influence Analysis
      3. Biology and Chemistry
         1. Molecular Structures
         2. Phylogenetic Trees
         3. Protein Interactions
         4. Metabolic Networks
      4. Logistics and Operations Research
         1. Transportation Networks
         2. Supply Chain Management
         3. Scheduling Problems
         4. Resource Allocation