Question

If G has n vertices and is regular of degree r, how many edges has G?

Answer 1

Answer:

We know that the sum of the degrees of the vertices of the graph is twice the number of edges of the graph.

Let $G$ be a graph with vertices $v_1,v_2,\cdots,v_n$

Then $\sum_{i=1}^ndeg(v_i)=2e$ where $e$ is the number of edges of the graph. Since the graph is regular of degree r then all vertices have degree r. Thus,

$\sum_{i=1}^n r=2e\\nr=2e\\e=\frac{nr}{2}$