Strong edge-magic graphs of maximum size

An edge-magic total labeling on G   is a one-to-one map λλ from V(G)∪E(G)V(G)∪E(G) onto the integers 1,2,…,|V(G)∪E(G)|1,2,…,|V(G)∪E(G)| with the property that, given any edge (x,y)(x,y), λ(x)+λ(x,y)+λ(y)=kλ(x)+λ(x,y)+λ(y)=k for some constant k. The labeling is strong   if all the smallest labels are assigned to the vertices. Enomoto et al. proved that a graph admitting a strong labeling can have at most 2|V(G)|-32|V(G)|-3 edges. In this paper we study graphs of this maximum size.