**Connectivity **is a basic concept of graph theory. It defines whether a graph is connected or disconnected. Without connectivity, it is not possible to traverse a graph from one vertex to another vertex.

• **A graph is said to be connected graph** if there is a path between every pair of vertex. From every vertex to any other vertex there must be some path to traverse. This is called the connectivity of a graph.

• **A graph is said to be disconnected**, if there exists multiple disconnected vertices and edges.

Graph connectivity theories are essential in network applications, routing transportation networks, network tolerance etc.

**Example of connected Graph:**

In the above example, it is possible to travel from one vertex to another vertex. Here, we can traverse from vertex B to H using the path B -> A -> D -> F -> E -> H. Hence it is a connected graph.

**Example of disconnected Graph:**

In the above example, it is not possible to traverse from vertex B to H because there is no path between them directly or indirectly. Hence, it is a disconnected graph.

## Walk:

A **walk **can be defined as a sequence of edges and vertices of a graph. When we have a graph and traverse it, then that traverse will be known as a walk.

In a walk, there can be repeated edges and vertices.

The number of edges which is covered in a walk will be known as the Length of the walk.

There are two types of the walk, which are described as follows:

**Open walkClosed walk**

For example: In this example, we have a graph, which is described as follows:

In the above graph, there can be many walks, but some of them are described as follows:

## Trail:

A trail is a walk in which no edge is repeated. The vertices can be repeated.

## Path:

A walk in which no vertices and edges are repeated is called a path.

So for a path, the following two points are important, which are described as follows:

Edges cannot be repeated

Vertex cannot be repeated

In the above graph, there is a path, which is described as follows:

## Circuit:

A closed trial which contains at least three edges is called a circuit.

So for a circuit, the following two points are important, which are described as follows:

Edges cannot be repeated

Vertex can be repeated

In the above graph, there is a circuit, which is described as follows:

## Cycle:

A cycle is a circuit in which no vertex is repeated other than the starting and ending vertices.