How to find whether it is possible for each vertex of a graph to have a different degree?

I want to prove whether it is possible for a graph to have different degrees for each vertex. I think that it can be possible with an example, but I can't prove it with mathematics.

If you are talking of simple graphs then clearly in any connected component containing n(>1) vertices the n vertex degrees will have degrees among the numbers $\{1,2,3\cdots n-1\}$ and so by the pigeonhole principle at least 2 vertices will have the same degree. The conclusion is false if we consider graphs with loops or with multiple edges.

Every simple graph with at least two vertices has at least two vertices of the same degree.

http://www.student.math.uwaterloo.ca/~math239/winter2008/t7sol.pdf

http://www.mymathforum.com/viewtopic.php?f=27&t=20418

This was among the first result when searching for "two vertices" "same degree". You can also find many books containing this claim.