Graph Theory
Graph Theory is a branch of mathematics concerned with the study of graphs. These graphs are not the data plots used for visualizing numerical data but are instead mathematical structures used to model pairwise relations between objects. Here's an in-depth look at Graph Theory:
History
Basic Concepts
- Vertices (Nodes): Represent objects or entities in the graph.
- Edges: Represent the relationships or connections between vertices. An edge can be directed or undirected, meaning it can specify a one-way or two-way relationship, respectively.
- Graph Types:
- Simple Graph - No loops (edges from a vertex to itself) or multiple edges between the same pair of vertices.
- Multigraph - Allows multiple edges between the same pair of vertices.
- Weighted Graph - Edges have weights or costs associated with them.
- Path: A sequence of edges that allows traversal from one vertex to another. A path is simple if no vertex is repeated.
- Connectedness: A graph is connected if there is a path between every pair of vertices.
- Graph Properties: Includes concepts like degree (number of edges incident to a vertex), connectivity, cycles, and so on.
Applications
- Computer Science: Used in algorithms for routing, network design, search engines, and social network analysis.
- Chemistry: Molecular structures are modeled as graphs where atoms are vertices, and bonds are edges.
- Transportation Networks: Graph theory helps in optimizing routes, traffic flow, and infrastructure planning.
- Social Sciences: Analysis of social networks, spread of information or diseases, etc.
Notable Theorems and Results
- Euler's Formula: For a connected planar graph, V - E + F = 2, where V, E, and F are the numbers of vertices, edges, and faces, respectively.
- Kuratowski's Theorem: A graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 or K3,3.
- Four Color Theorem: Any map in a plane can be colored with four colors such that no two adjacent regions share the same color.
External Resources
Related Topics