Degree (Graph Theory)

In the mathematical field of graph theory the degree or valency of a vertex v is the number of edges incident to v (with loops being counted twice). We write \deg(v) to denote the degree of v. In a directed graph the indegree of a vertex v is the number of edges terminating at v and the outdegree is the number of edges originating at v. We write \deg^+(v) and \deg^-(v) to denote the indegree and outdegree of v. A vertex with \deg(v)=0 is called isolated. A vertex with \deg(v)=1 is called a leaf. 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. A vertex with \deg^+(v)=0 is called a source and a vertex with \deg^-(v)=0 is called a sink.

Some theorems

Given a directed graph G for each vertex v of G
\deg(v) = \deg^+(v) + \deg^-(v)
The number of vertices with odd degree in any graph is even Given a graph G=(V,E) then
\sum_{v \in V} \deg(v) = 2|E|

 

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