Discrete Mathematics : In-Degree & Out-Degree of a Vertex

In this section of **In-Degree & Out-Degree of a Vertex**, we cover the chapter "Graph Theory". Here we explain how to find the indegree and out degree of a vertex in a graph.

**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 on-degree of a vertex, just count the number of edges starting from the vertex.

Therefore **the degree of a vertex** is equal to the sum of **In-Degree of a vertex** and the **Out-Degree of a vertex** i.e.

Deg(v) = deg^{−}(v) + deg^{+}(v)

**Example**: Find the In -Degree, Out-degree, and degree of each vertex of a graph given below.

**In-Degree**and

**Out-Degree**of each vertex in a graph is

In-Degree of a vertex'v_{1}' = deg(v_{1}) = 1 andOut-Degree of a vertex'v_{1}' = deg(v_{1}) = 2

In-Degree of a vertex'v_{2}' = deg(v_{2}) = 1 andOut-Degree of a vertex'v_{2}' = deg(v_{2}) = 3

In-Degree of a vertex'v_{3}' = deg(v_{3}) = 1 andOut-Degree of a vertex'v_{3}' = deg(v_{3}) = 2

In-Degree of a vertex'v_{4}' = deg(v_{4}) = 5 andOut-Degree of a vertex'v_{4}' = deg(v_{4}) = 0

In-Degree of a vertex'v_{5}' = deg(v_{5}) = 1 andOut-Degree of a vertex'v_{5}' = deg(v_{5}) = 2

In-Degree of a vertex'v_{6}' = deg(v_{6}) = 0 andOut-Degree of a vertex'v_{6}' = deg(v_{6}) = 0

And by the definition, the degree of a vertex is

Deg(v) = deg. Therefore,^{−}(v) + deg^{+}(v)

Degree of a vertex'v_{1}' = deg(v_{1}) = 1 + 2 = 3

Degree of a vertex'v_{2}' = deg(v_{2}) = 1 + 3 = 4

Degree of a vertex'v_{3}' = deg(v_{3}) = 1 + 2 = 3

Degree of a vertex'v_{4}' = deg(v_{4}) = 5 + 0 = 5

Degree of a vertex'v_{5}' = deg(v_{5}) = 1 + 2 = 3

Degree of a vertex'v_{6}' = deg(v_{6}) = 0 + 0 = 0

**Try it : ** Find the In -Degree, Out-degree, and degree of each vertex of a graph given below.

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