Graph Theory Tutorial

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 that connect pairs of vertices.

Basic Concepts

Vertices: These are the individual objects that are connected by edges.
Edges: These are the connections between vertices.

Types of Graphs

Directed Graph: Edges have a direction, indicating a one-way relationship.
Undirected Graph: Edges do not have a direction, indicating a two-way relationship.

Applications

Graph theory has many applications in various fields, including:

Computer Science: Network flow, shortest paths, and social networks.
Operations Research: Scheduling, network design, and logistics.
Physics: Modeling the interactions between particles.

Key Definitions

Degree: The number of edges connected to a vertex.
Path: A sequence of vertices connected by edges.
Cycle: A path that starts and ends at the same vertex.

Example

To understand these concepts better, consider the following example of a social network:

Vertices: Individuals in the network.
Edges: Friendships between individuals.

Image

Here is an image of a simple graph to illustrate these concepts: