Explain the concept of vertices and edges in a graph.

In graph theory, a graph is a mathematical structure that consists of two fundamental components: vertices and edges.

Vertices, also known as nodes, are the fundamental building blocks of a graph. They represent the entities or objects within the graph. For example, in a social network graph, vertices can represent individuals, while in a transportation network graph, vertices can represent cities or locations.

Edges, on the other hand, are the connections or relationships between vertices. They represent the interactions, associations, or dependencies between the entities represented by the vertices. Edges are typically represented by lines or arcs connecting the vertices. For instance, in a social network graph, edges can represent friendships or connections between individuals, while in a transportation network graph, edges can represent roads or routes connecting cities.

The combination of vertices and edges forms the structure of a graph, allowing us to visualize and analyze relationships between different entities. Graphs can be used to model a wide range of real-world scenarios, such as social networks, computer networks, biological networks, and many more. The study of graphs and their properties is known as graph theory, which plays a crucial role in various fields including computer science, mathematics, and operations research.