Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (or nodes) and edges. The vertices represent entities, and the edges represent relationships between the entities.

Basic Concepts

  • Vertices: The individual objects in a graph.
  • Edges: The connections between vertices.
  • Path: A sequence of edges that connects a sequence of vertices.
  • Cycle: A path that starts and ends at the same vertex.
  • Tree: A connected acyclic graph.

Types of Graphs

  • Directed Graph: Edges have a direction.
  • Undirected Graph: Edges have no direction.
  • Weighted Graph: Edges have associated weights.

Applications

Graph theory has applications in various fields, including:

  • Computer Science: Network design, social networks, algorithms.
  • Physics: Modeling complex systems, such as the internet and social networks.
  • Engineering: Circuit design, network design.
  • Mathematics: Study of algebraic structures.

Example

To understand graph theory better, consider the following example:

  • Vertices: Cities
  • Edges: Roads connecting the cities

You can visualize this as a graph where cities are vertices and roads are edges.

Further Reading

For more information on graph theory, you can visit our Graph Theory Tutorial.

Graph Theory Example