Let $v_k$ be the number of vertices of degree $k$, V and E the set of vertices and edges respectively. Then $|V| = \sum_k v_k$.
I have to show the double counting formula for a closed surface: $$ \sum_k kv_k = 2|E| $$
I would like to give a rigorous proof. I was searching on the internet and I found proofs that explain the intuition.
I was thinking about and I got something like this:
$$ \sum_{k} k v_k =\sum_{v_i \in V} d_i = \sum_{v_i \in V} \sum_{(v_i v_k) \in E} 1 \stackrel{(1)}{=} \sum_{(v_i v_k) \in E} \sum_{ v \in \{v_i, v_k \}} 1 = \sum_{(v_i v_k) \in E} 2 = 2|E| $$
but I am not sure if this is correct, especially in the step (1). Note that $d_i$ is the degree of $v_i$.
- How could I improve/correct this proof?
- What would be the correct proof? Or where could I find the correct one?