Question

A graph G and its complement G have 8 and 7 edges respectively. What is the
number of vertices in G ?

Answer 1

Let's consider that number of this full graph vertices is x. Than, the number of edges is the product of x and x-1, and this product has to be divided by 2, because the graph is not oriented. Also, the number of edges is 7+8=15 by the problem statement. Thus, we have an equation:

x(x-1)/2 = 15 -> x(x-1)=30

The roots of this equation are 6 and -5, but in order to x is the number of vertices , x >=0, so x is 6.

Answer. 6 vertices.