The Hand Shaking Lemma

Question

In any graph G=(V,E)

[the hand shaking lemma]

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

(original at https://i.stack.imgur.com/af4en.png)

where |E| donetes the number of edges

I alredy tried to count to edges.For a more normal argument i used the induction on the number of edges but i came croos a problem.I could not keep doing the inductiun

Answer 1

Denote by $d(v)$ the degree of a vertex $v$ in $G$. Then $$\sum_{v\in V}d(v)$$ counts the sum of all the edges of $G$ twice, once for each vertex. Thus the result follows.

Answer 2

There are two ways to count the total number of degrees:

1. sum up the degrees of all vertices (the left hand side of the equation)
2. note that each edge is connected to two vertices, so, each edge adds 2 to the total sum of degrees of a graph (the right hand side of the equation)