Number of edges in degree-magic graphs

A graph GG is called degree-magic if it admits a labelling of the edges by integers 1,2,…,|E(G)|1,2,…,|E(G)| such that the sum of the labels of the edges incident with any vertex vv is equal to 12(1+|E(G)|)deg(v). Degree-magic graphs extend supermagic regular graphs. In this paper, we present some bounds on the number of edges in degree-magic graphs.