Consider any complete bipartite graph $K_{p,q}$. Express the number of edges in $K_{p,q}^C$, the complement of $K_{p,q}$, as a function of $n$, the total number of verticies.
Now, I know that I could do this by subtracting the number of edges in $K_{p,q}$ by the number of edges in $K_n$, a complete graph with $n$ vertices. However, I do not see how it is possible to determine the number of edges in $K_{p,q}$ merely as a function of $n$ because I need to know how many verticies are on each side of the graph in order to determine this. For instance, $K_{1,3}$ and $K_{2,2}$ both have 4 verticies, however, the former one has 3 edges whereas the latter has 4 edges because $$ The~number ~of ~edges ~in ~a ~complete ~bipartite ~graph~ K_{p,q} = pq$$
So I'm not sure that this can really be expressed solely as a function of $n$
Best Answer
So for those who are curious, the question was meant to be interpreted as follows.
Maximize the number of edges in the graph $G^C$ which has $n$ vertices.
We know that the number of edges in a complete graph, $K_n$ is $\frac{n(n-1)}{2}$ and the minimum number of edges in a connected graph of $n$ vertices is $n-1$. So that means that means $$ max\{|E_{G^C}|\} = \frac{n(n-1)}{2} - (n-1) $$ $$ s.t. n=p+q$$
Doing some algebra, we can arrive at $$ max\{|E_{G^C}|\} = \frac{(n-1)(n-2)}{2} $$ and this occurs when $p = 1$ and $ q = n-1$ or vice-versa