The study of the possible valences for edge-magic labelings of graphs has motivated us to introduce the concept of perfect edge-magic graphs. Intuitively speaking, an edge-magic graph is perfect edge-magic if all possible theoretical valences occur. In particular, we prove that for each integer m > 0, that is the power of an odd prime, and for each natural number n, the crown product C-m circle dot (K-n) over bar is perfect edge-magic. Related results are also provided concerning other families of unicyclic graphs. Furthermore, several open questions that suggest interesting lines for future research are also proposed.