**What Is Multigraph in Data Structure?**

When studying data structures, it is important to understand different types of graphs. One such type is a multigraph, which is a graph that allows multiple edges between two vertices and even loops where an edge connects a vertex to itself.

__Definition:__

A multigraph, also known as a pseudograph, is an extension of a simple graph. In a simple graph, each edge connects two distinct vertices, whereas in a multigraph, there can be multiple edges between the same pair of vertices.

__Representation:__

In order to represent a multigraph efficiently, we can use an adjacency list or an adjacency matrix. The choice depends on the specific requirements and operations performed on the multigraph.

## Adjacency List Representation:

In the adjacency list representation of a multigraph, we maintain an array of lists. Each element in the array represents a vertex, and the corresponding list contains all the adjacent vertices connected by edges.

Here is an example of an adjacency list representation of a multigraph:

- List 0: 1 -> 2 -> 2 -> 3
- List 1: 0 -> 2 -> null
- List 2: 0 -> 0 -> 1
- List 3: 0

## Adjacency Matrix Representation:

In the adjacency matrix representation of a multigraph, we use a two-dimensional matrix to represent the connections between vertices. The matrix element at index [i][j] represents the number of edges between vertex i and vertex j.

Here is an example of an adjacency matrix representation of a multigraph:

- 0 1 2 3
- 0 0 1 2 0
- 1 1 0 0 0
- 2 2 0 0 1
- 3 0 0 1 0

### Operations on Multigraph:

Various operations can be performed on a multigraph, including:

**AddVertex(V):**Adds a new vertex V to the multigraph.**AddEdge(V, W):**Adds a new edge between vertices V and W in the multigraph.**DeleteVertex(V):**Deletes the vertex V from the multigraph.**DeleteEdge(V, W):**Deletes the edge between vertices V and W from the multigraph.**GetAdjacentVertices(V):**Returns a list of all vertices adjacent to vertex V in the multigraph.**GetDegree(V):**Returns the degree of vertex V, i.e., the number of edges incident to it in the multigraph.

__Applications:__

Multigraphs find applications in various fields, including network analysis, transportation systems, social networks, and computer science algorithms. They provide a more flexible representation when multiple connections between vertices are required.

In conclusion, understanding multigraphs is crucial for analyzing complex relationships and connectivity patterns in real-world scenarios. By using appropriate data structures and algorithms, we can efficiently represent and manipulate multigraphs to solve a wide range of problems.