Degree

The degree of a node in a graph is the number of edges that are incident to the node. It is generally denoted as $deg(v)$ .

Properties

In a complete graph, every node has the maximum possible degree, $n-1$. Therefore, a complete graph has $n(n-1)$ edges.

Handshaking Lemma

The handshaking lemma states that for a given graph $G$,

$$\sum _{v\in V}deg(v)=2|E|$$

In other words, the sum of degrees of all vertices is equal to twice the amount of edges. Since an edge connects two nodes, a single edge contributes twice to the sum of degrees of the graph, is one way of thinking about it.

This lemma also results in the fact that in any undirected graph, the number of vertices with an odd degree is even.