If X has an n-fold universal cover K and is connected then the order of its fundamental group is less than or equal to n.
I realized this when noting the fact that any path from a point x in X to itself can be lifted to a path in K from a fixed point x' in the preimage of x to another point in the preimage (under the cover map). This can be done for every path and then we realize that two such liftings are homotopic relative to their endpoints iff their endpoints are equal. Then we have atleast n classes of liftings since the cover is n-fold. Done.
This is an application of fundamental graph theory. This also got me thinking, what are some other fun applications of graph theory to topology. Are there any applications that go the other way?