How are vertices and edges represented in a graph data structure?

In a graph data structure, vertices are represented as nodes and edges are represented as connections between these nodes.

A graph data structure is a non-linear data structure that represents a pictorial structure of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges.

The representation of vertices and edges in a graph data structure can be done in two ways: adjacency matrix and adjacency list. In an adjacency matrix, a 2D array of size V x V (where V is the number of vertices) is used. Each row and column represent a vertex. If the value of any element a[i][j] is 1, it represents that there is an edge connecting vertex i and vertex j.

On the other hand, an adjacency list represents a graph as an array of linked lists. The index of the array represents a vertex and each element in its linked list represents the other vertices that form an edge with the vertex. The adjacency list is preferred over the adjacency matrix for sparse graphs as it takes less space.

In object-oriented programming, a vertex can be represented as an object of a Vertex class, which may contain properties like the vertex name, its neighbours, its state (visited or not visited), and other relevant information. An edge can be represented as an object of an Edge class, which may contain properties like the vertices it connects, its weight (if it's a weighted graph), and other relevant information.

In summary, vertices and edges in a graph data structure are abstractly represented as nodes and connections between these nodes. The concrete implementation can vary, with common methods being adjacency matrices and adjacency lists, or using object-oriented programming concepts.