Discrete Mathematics

Degree of a vertex in a directed graph
In case of Directed Graph, the in-degree and out-degree and degree of a vertex can be determined by the following steps.
In-degree of a vertex
The In-Degree of a vertex v written by deg^-(v), is the number of edges with v as the terminated vertex. To find the in-degree of a vertex, just count the number of edges ends at the vertex.
Out-degree of a vertex
The Out-Degree of a vertex V written by deg⁺ (v), is the number of edges with v as the initial vertex. To find the out-degree of a vertex, just count the number of edges starting from the vertex. Thus degree of a vertex is equal to the sum of In-Degree of a Vertex and Out-Degree of a Vertex i.e.
Deg (v) = deg⁻ (v) + deg⁺(v)

Example: Find the degree of each vertex of a graph given below.

The In-Degree and Out-Degree of each vertex in a graph is
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

And by the definition, the degree of a vertex is Deg(v) = deg−(v) + deg+(v). Therefore,

Degree of a vertex 'v1' is deg(v1) = 1 + 2 = 3
Degree of a vertex 'v2' is deg(v2) = 1 + 3 = 4
Degree of a vertex 'v3' is deg(v3) = 1 + 2 = 3
Degree of a vertex 'v4' is deg(v4) = 5 + 0 = 5
Degree of a vertex 'v5' is deg(v5) = 1 + 2 = 3
Degree of a vertex 'v6' is deg(v6) = 0 + 0 = 0
Degree sequence of a graph
When we find the degree of each vertex in a graph, we just write the degree in the ascending order then this sequence is known as the degree sequence of a graph. For Example, graphs having the degree sequence is 3 for the number of vertices in a graph is 3, 4 and 5 is

Degree Sequence is <2, 2, 2>

Degree Sequence is <2, 2, 2, 2>

Degree Sequence is <2, 2, 2, 2, 2>