A graph is total edge-magic if there exists a bijection such that for each , we have as a constant. For a graph , denote the set of all total edge-magic labelings. The magic strength of is the minimum of all constants among all labelings in , denoted by . The maximum of all constants among is called the maximum magic strength of and denoted by .
Hegde and Shetty classify a magic graph as strong if , ideal magic if , and \textbf{weak magic} if . A total edge-magic graph is called a super edge-magic if . The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.