Degree (graph theory)
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- This article is about the term "degree" as used in graph theory. For alternate meanings, see degree (mathematics) or degree.
In graph theory, the degree (or valency) of a vertex is the number of edges incident to the vertex. The degree of a vertex v is denoted deg(v).
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[edit] Undirected graphs
For an undirected graph, the degree of a vertex is the number of edges incident to the vertex. This means that each loop is counted twice. This is because each edge has two endpoints and each endpoint adds to the degree.
The graph shown to the right has the following degrees:
| Vertex | Degree |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 2 |
| 4 | 3 |
| 5 | 3 |
| 6 | 1 |
[edit] Directed graphs
In a directed graph, an edge has two distinct ends: a head (the end with an arrow) and a tail. Each end is counted separately. The sum of head endpoints count toward the indegree and the sum of tail endpoints count toward the outdegree.
The indegree is denoted deg + (v) and the outdegree as deg − (v)
The image to the right has the following degrees:
| Vertex | Indegree | Outdegree |
|---|---|---|
| 1 | 0 | 2 |
| 2 | 2 | 0 |
| 3 | 2 | 2 |
| 4 | 1 | 1 |
[edit] Special cases of degree value
[edit] Isolated vertex
If a vertex has a zero degree, it is called an isolated vertex.
[edit] Leaf vertex
If a vertex has a unity degree, it is called a leaf. This is fairly common in trees in graph theory and trees in data structures.
[edit] Regular graph
If each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k.
[edit] Source
A vertex with deg + (v) = 0 is called a source. This name comes from the fact that the node is the origin/source of all of its incident edges. In the image under Directed graphs, vertex 1 is a source node.
[edit] Sink
A vertex with deg − (v) = 0 is called a sink. Similarly to source nodes, a sink gets its name from the fact that the node is the termination/destination/sink of all of its incident edges. In the image under Directed graphs, vertex 2 is a sink node.
[edit] Eulerian graph
A graph is Eulerian if and only if every vertex is of even degree.
[edit] Some theorems
The degree sum formula states that, given a graph G = (V,E),
since each edge is incident to two vertices. The formula implies that in any graph, the number of vertices with odd degree is even.
