Question

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

Answer 1

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=(nr)/(2)`