Prove that in a graph G, the sum of the degrees of the vertices is equal to twice the number of edges. Consequently, the number of vertices with odd degree is even

Let G be a graph with e edges and n vertices v1,v2,v3,...,vn.

Since each edge is incident on two vertices, it contributes 2 to the sum of degree of vertices in graph G. Thus the sum of degrees of all vertices in G is twice the number of edges in G:$∑^n_{i=1}degree(v_i)=2e$

Let the degrees of first r vertices be even and the remaining (n−r) vertices have odd degrees, then:

$∑^n_{i=1}degree(v_i)=∑^r_{i=1}degree(v_i)+∑^n_{i=r+1}degree(v_i)$

$\implies ∑^n_{i=1}degree(v_i)+∑^r_{i=1}degree(v_i)$

$\implies ∑^n_{i=r+1}degree(v_i)$ is even.

But, the for $i=r+1,r+2,...,n$ each $d(v_i)$ is odd. So, the number of terms in

$∑^n_{i=r+1}degree(v_i)$ must be even. So, $(n-r)$ is even.