What is a Graph?
Welcome to the world of Graph Theory! Let’s start with the most fundamental question: What exactly is a graph?
In mathematics, a graph is an abstract structure used to model relationships between objects. It consists of two primary components:
Vertices (also called nodes): These are the fundamental entities or objects within the graph. Think of them as points.
Edges (also called links or arcs): These are the connections that link pairs of vertices, representing the relationship between them.
Formally, a graph \(G\) is defined as an ordered pair \(G = (V, E)\), where:
- \(V = \{v_1, v_2, …, v_n\}\) is the set of vertices
- \(E = \{(v_i, v_j) | v_i, v_j \in V\}\) is the set of edges
If an edge connects two vertices \(v_1\) and \(v_2\), then we denote the edge by \(v_1v_2\), which is same as \(v_2v_1\). Two vertices are said to be adjacent if they are connected by an edge.
For instance, shown above are several ways of drawing the same graph. Notice that all three have the same structure of a string of five vertices connected in a row.
Example: A Simple Social Network
One of the most intuitive examples of a graph is a social network. Let’s model a small group of friends.
- Vertices (\(V\)): Each person is a vertex. Let’s take four friends: Alice, Bob, Charlie, and Diana. So,
V = {Alice, Bob, Charlie, Diana}. - Edges (\(E\)): Each friendship is an edge connecting two people.
- Alice is friends with Bob.
- Alice is also friends with Charlie.
- Bob and Charlie are friends.
- Charlie is friends with Diana.
The set of edges is E = {{Alice, Bob}, {Alice, Charlie}, {Bob, Charlie}, {Charlie, Diana}}.
Here is an interactive representation of this social network. You can drag the nodes around to see how they are connected.
This simple visualization reveals the structure of the group’s friendships. For example, we can quickly see that Alice and Diana are not directly connected but are linked through Charlie.
Why Do Graphs Matter?
Graphs are more than just a mathematical curiosity; they are a powerful tool for modeling the world around us. They provide a common language and a set of powerful algorithms to solve problems in:
- Computer Science: Network routing, database design, compiler optimization, and AI.
- Biology: Modeling protein-protein interactions.
- Social Sciences: Analyzing the spread of information and influence, modeling social networks and relationships.
- Logistics & Transportation: Optimizing routes, traffic flow analysis
- Telecommunications: Network design and optimization
- Operations Research: Scheduling, resource allocation
By understanding the basics of graphs, you unlock a new way to see and solve complex problems.