Degree of Vertex of a Graph

The degree of a vertex V is the number of edges incident with (connected to) that vertex.

Notation − deg(V)

In a simple graph with n vertices, the maximum degree of any vertex is −

deg(v) ≤ n - 1,  for all v in G

A vertex can form an edge with all other vertices except itself. So the degree of a vertex will be at most the number of vertices minus 1. If there is a loop at any vertex, then it is not a simple graph.

Degree of a vertex can be considered under two cases −

• Undirected Graph − Degree is the total number of edges at a vertex.
• Directed Graph − Each vertex has an indegree (incoming edges) and an outdegree (outgoing edges).

Degree of Vertex in an Undirected Graph

An undirected graph has no directed edges. The degree of a vertex is simply the count of edges connected to it.

Example 1

Take a look at the following graph −

In the above undirected graph −

• deg(a) = 2, as there are 2 edges meeting at vertex 'a'.
• deg(b) = 3, as there are 3 edges meeting at vertex 'b'.
• deg(c) = 1, as there is 1 edge at vertex 'c'. So 'c' is a pendent vertex.
• deg(d) = 2, as there are 2 edges meeting at vertex 'd'.
• deg(e) = 0, as there are 0 edges at vertex 'e'. So 'e' is an isolated vertex.

Example 2

Take a look at the following graph −

In the above graph, deg(a) = 2, deg(b) = 2, deg(c) = 2, deg(d) = 2, and deg(e) = 0. The vertex 'e' is an isolated vertex. The graph has no pendent vertices.

Degree of Vertex in a Directed Graph

In a directed graph, each vertex has an indegree and an outdegree.

Indegree − The number of edges coming into the vertex. Notation: deg⁻(V).

Outdegree − The number of edges going out from the vertex. Notation: deg⁺(V).

Example 1

Take a look at the following directed graph. Vertex 'a' has two edges 'ad' and 'ab' going outwards (outdegree = 2), and one edge 'ga' coming inwards (indegree = 1) −

The indegree and outdegree of all vertices are −

Example 2

Take a look at the following directed graph −

The indegree and outdegree of all vertices are −

Conclusion

The degree of a vertex counts the number of edges connected to it. In undirected graphs, vertices with degree 0 are isolated and those with degree 1 are pendent. In directed graphs, the degree is split into indegree (incoming edges) and outdegree (outgoing edges).