"Determine how many complete bipartite graphs have n vertices."

Question

Wouldn't this simply be an infinite number of graphs?

Answer 1

Well, not quite. You can only have so many graphs.

Let's start with $2$ vertices. The only corresponding complete bipartite would be $K_{1,1}$.

Then, for $3$ vertices, we'd only have $K_{1,2}$.

Then for $4$ vertices, we have both $K_{1,3}$ and $K_{2,2}$.

For $21$ vertices, we have all the way from $K_{1,20}$ to $K_{10,11}$.

In general, there are $\large \displaystyle \lfloor \frac{n}{2}\rfloor$ graphs.