  The super edge-magicness of graphs of equal order and size has been shown to
be important since such graphs can be used as seeds to answer many questions
related to (super) edge-magic labelings and other types of well studied
labelings, as for instance harmonious labelings. Also other questions related
to the area of combinatorics can be attacked and understood from the point of
view of super edge-magic graphs of equal order and size. For instance, the
design of Steiner triple systems, the study of the set of dual shuffle primes
and the Jacobsthal numbers. In this paper, we study the super edge-magic
properties of some types of super edge-magic graphs of equal order and size,
with the hope that they can be used later in the study of other related
questions. The negative results found in last section are specially interesting
since these kind of results are not common in the literature. Furthermore, the
few results found in this direction usually meet one of the following reasons:
too many vertices compared with the number of edges; too many edges compared
with the number of vertices; or parity conditions. In this case, all previous
reasons fail in our results.
