Graph Theory question? Assuming that friendship is always mutual, prove that in any group of n  2 persons, there are at least 2 persons with the same number of friends in the group.

How do I answer this question with a graph ?
You can view this as a question about simple graphs, but it will boil down to the pigeon hole principle. Suppose everyone has a different number of friends, so listing those numbers in order yields $\{ 1,...,n-1\}$ or $\{ 0,1,...,n-2\}$ (depending on what kind of assumptions you want to make). Regardless, you have $n-1$ possibilities with $n$ people.
This is already on the internet here:

http://puzzlesland.com/mathprob/graph-samedeg.html