In-degree and Out-degree in discrete mathematics

To understand the in-degree and out-degree of a vertex, we have to first learn about the concept of degree of a vertex. After that, we can easily understand the in-degree and out-degree of a vertex. We should know that the in-degree and out-degree can only be determined in the directed graph. We can calculate the degree of a vertex with the help of an undirected graph. In the undirected graph, we cannot calculate the in-degree and out-degree of a vertex.

Degree of a vertex

If we want to find the degree of each vertex in a graph, in this case, we have to count the number of relations that are established by a particular vertex with the other vertex. In other words, we can determine the degree of a vertex with the help of calculating the number of edges connecting to that vertex. The degree of a vertex is indicated with the help of deg(v). If there is a simple graph, which contains n number of vertices, in this case, the degree of any vertex will be:

A vertex has the ability to form an edge with all other vertices in a graph except by itself. So in a simple graph, the degree of a vertex will find out by the number of vertices in a graph minus 1. Here 1 is used for the self vertex because it does not make a loop by itself. If the graph contains the vertices which have the self-loop, then that type of graph will not be a simple graph.

Example:

In this example, we have a graph that has 6 vertices, i.e., a, b, c, d, e, and f. The vertex 'a' has degree 5, and all the other vertices have a degree 1. If any vertex has degree 1, then that type of vertex will be known as the 'end vertex'.

In-degree and Out-degree in discrete mathematics

There are two cases of graphs in which we can consider the degree of a vertex, which are described as follows:

Undirected graph
Directed graph

Now we will learn the degree of a vertex in a directed graph and the degree of a vertex in an undirected graph in detail.

Degree of a vertex in an Undirected graph

If there is an undirected graph, then in this type of graph, there will be no directed edge. The examples to determine the degree of a vertex in an undirected graph are described as follows:

Example 1: In this example, we will consider an undirected graph. Now we will find out the degree of each vertex in that graph.

Solution: In the above-undirected graph, there are total 5 numbers of vertices, i.e., a, b, c, d, and e. The degree of each vertex is described as follows:

The above graph contains 2 edges, which meet at vertex 'a'. Hence Deg(a) = 2
This graph contains 3 edges, which meet at vertex 'b'. Hence Deg(b) = 3
The above graph contains 1 edge, which meet at vertex 'c'. Hence Deg(c) = 1. The vertex c is also known as the pendent vertex.
The above graph contains 2 edges, which meet at vertex 'd'. Hence Deg(d) = 2.
The above graph contains 0 edges, which meet at vertex 'e'. Hence Deg(a) = 0. The vertex e can also be called the isolated vertex.

Example 2: In this example, we will consider an undirected graph. Now we will find out the degree of each vertex in that graph.

Solution: In the above undirected graph, there are total 5 numbers of vertices, i.e., a, b, c, d, and e. The degree of each vertex is described as follows:

Degree of vertex a = deg(a) = 2

Degree of vertex b = deg(b) = 2

Degree of vertex c = deg(c) = 2

Degree of vertex d = deg(d) = 2

Degree of vertex e = deg(e) = 0

In this graph, there is no pendent vertex, and vertex 'e' is an isolated vertex.

Degree of vertex in Directed graph

If the graph is a directed graph, then in this graph, each vertex must have an in-degree and out-degree. Suppose there is a directed graph. In this graph, we can use the following steps to find out the in-degree, out-degree, and degree of a vertex.

In-degree of a vertex

The in-degree of a vertex can be described as a number of edges with v, where v is used to indicate the terminal vertex. In other words, we can describe it as a number of edges coming to the vertex. With the help of syntax deg^-(v), we can write the in-degree of a vertex. If we want to determine the in-degree of a vertex, for this, we have to count the number of edges that ends at the vertex.

Out-degree of a vertex

The out-degree of a vertex can be described as a number of edges with v, where v is used to indicate the initial vertex. In other words, we can describe it as a number of edges coming out from the vertex. With the help of syntax deg⁺(v), we can write the out-degree of a vertex. If we want to determine the out-degree of a vertex, for this, we have to count the number of edges that begins from the vertex.

Degree of a vertex

The degree of a vertex is indicated with the help of deg(v), which is equal to the addition of in-degree of a vertex and out-degree of a vertex. The symbolic representation of degree of a vertex is described as follows:

Example 1: In this example, we have a graph, and we have to determine the degree of each vertex.

Solution: For this, we will first find out the degree of a vertex, in-degree of a vertex and then the out-degree of a vertex.

As we can see that the above graph contains the total 6 vertex, i.e., v1, v2, v3, v4, v5 and v6.

In-degree:

In-degree of a vertex v1 = deg(v1) = 1

In-degree of a vertex v2 = deg(v2) = 1

In-degree of a vertex v3 = deg(v3) = 1

In-degree of a vertex v4 = deg(v4) = 5

In-degree of a vertex v5 = deg(v5) = 1

In-degree of a vertex v6 = deg(v6) = 0

Out-degree:

Out-degree of a vertex v1 = deg(v1) = 2

Out-degree of a vertex v2 = deg(v2) = 3

Out-degree of a vertex v3 = deg(v3) = 2

Out-degree of a vertex v4 = deg(v4) = 0

Out-degree of a vertex v5 = deg(v5) = 2

Out-degree of a vertex v6 = deg(v6) = 0

Degree of a vertex

With the help of the definition described above, we know that the degree of a vertex Deg(v) = deg^-(v) + deg⁺(v). Now we will calculate it with the help of this formula like this:

Degree of a vertex v1 = deg(v1) = 1+2 = 3

Degree of a vertex v2 = deg(v2) = 1+3 = 4

Degree of a vertex v3 = deg(v3) = 1+2 = 3

Degree of a vertex v4 = deg(v4) = 5+0 = 5

Degree of a vertex v5 = deg(v5) = 1+2 = 3

Degree of a vertex v6 = deg(v6) = 0+0 = 0

Example 2:

In this example, we have a directed graph with 7 vertices. The vertex 'a' contains 2 edges, i.e., 'ad' and 'ab', which are going outwards. Hence, vertex 'a' contains the out-degree, which is 2. Similarly, the vertex 'a' also has an edge 'ga', which is coming toward this vertex 'a'. Hence, the vertex 'a' contains the in-degree, which is 1.

Solution: The in-degree and out-degree of all the above vertices are described as follows:

In-degree:

In-degree of a vertex a = deg(a) = 1

In-degree of a vertex b = deg(b) = 2

In-degree of a vertex c = deg(c) = 2

In-degree of a vertex d = deg(d) = 1

In-degree of a vertex e = deg(e) = 1

In-degree of a vertex f = deg(f) = 1

In-degree of a vertex g = deg(g) = 0

Out-degree:

Out-degree of a vertex a = deg(a) = 2

Out-degree of a vertex b = deg(b) = 0

Out-degree of a vertex c = deg(c) = 1

Out-degree of a vertex d = deg(d) = 1

Out-degree of a vertex e = deg(e) = 1

Out-degree of a vertex f = deg(f) = 1

Out-degree of a vertex g = deg(g) = 2

Degree of each vertex:

We known that the degree of a vertex Deg(v) = deg^-(v) + deg⁺(v). Now we will calculate it with the help of this formula like this:

Degree of a vertex a = deg(a) = 1+2 = 3

Degree of a vertex b = deg(b) = 2+0 = 2

Degree of a vertex c = deg(c) = 2+1 = 3

Degree of a vertex d = deg(d) = 1+1 = 2

Degree of a vertex e = deg(e) = 1+1 = 2

Degree of a vertex f = deg(f) = 1+1 = 2

Degree of a vertex g = deg(g) = 0+2 = 2

Example 3: In this example, we have a directed graph with 5 vertices. The vertex 'a' contains 1 edge, i.e., 'ae', which are going outwards. Hence, vertex 'a' contains an out-degree, which is 1. Similarly, the vertex 'a' also has an edge 'ba', which is coming toward this vertex 'a'. Hence, the vertex 'a' contains the in-degree, which is 1.

Solution: The in-degree and out-degree of all the above vertices are described as follows:

In-degree

In-degree of a vertex a = deg(a) = 1

In-degree of a vertex b = deg(b) = 0

In-degree of a vertex c = deg(c) = 2

In-degree of a vertex d = deg(d) = 1

In-degree of a vertex e = deg(e) = 1

Out-degree:

Out-degree of a vertex a = deg(a) = 1

Out-degree of a vertex b = deg(b) = 2

Out-degree of a vertex c = deg(c) = 0

Out-degree of a vertex d = deg(d) = 1

Out-degree of a vertex e = deg(e) = 1

Degree of each vertex:

We known that the degree of a vertex Deg(v) = deg^-(v) + deg⁺(v). Now we will calculate it with the help of this formula like this:

Degree of a vertex a = deg(a) = 1+1 = 2

Degree of a vertex b = deg(b) = 0+2 = 2

Degree of a vertex c = deg(c) = 2+0 = 2

Degree of a vertex d = deg(d) = 1+1 = 2

Degree of a vertex e = deg(e) = 1+1 = 2

Example 4: In this example, we have a graph, and we have to determine the degree, in-degree, and out-degree of each vertex.

Solution: For this, we will first find out the in-degree of a vertex and then the out-degree of a vertex.

As we can see that the above graph contains the total 8 vertex, i.e., 0, 1, 2, 3, 4, 5, and 6.

In-degree:

In-degree of a vertex 0 = deg(0) = 1

In-degree of a vertex 1 = deg(1) = 2

In-degree of a vertex 2 = deg(2) = 2

In-degree of a vertex 3 = deg(3) = 2

In-degree of a vertex 4 = deg(4) = 2

In-degree of a vertex 5 = deg(5) = 2

In-degree of a vertex 6 = deg(6) = 2

Out-degree:

Out-degree of a vertex 0 = deg(0) = 2

Out-degree of a vertex 1 = deg(1) = 1

Out-degree of a vertex 2 = deg(2) = 3

Out-degree of a vertex 3 = deg(3) = 2

Out-degree of a vertex 4 = deg(4) = 2

Out-degree of a vertex 5 = deg(5) = 2

Out-degree of a vertex 6 = deg(6) = 1

Degree of each vertex:

We known that the degree of a vertex Deg(v) = deg^-(v) + deg⁺(v). Now we will calculate it with the help of this formula like this:

Degree of a vertex 0 = deg(0) = 1+2 = 3

Degree of a vertex 1 = deg(1) = 2+1 = 3

Degree of a vertex 2 = deg(2) = 2+3 = 5

Degree of a vertex 3 = deg(3) = 2+2 = 4

Degree of a vertex 4 = deg(4) = 2+2 = 4

Degree of a vertex 5 = deg(5) = 2+2 = 4

Degree of a vertex 6 = deg(5) = 2+1 = 3

Degree sequence of a Graph

To determine the degree sequence of a graph, we have to first determine the degree of each vertex in a graph. After that, we will write these degrees in ascending order. This order/sequence can be called the degree sequence of a graph.

For example: In this example, we have three graphs that have 3, 4, and 5 vertices, and the degree sequence of all the graphs is 3.