On magic graphs

A( p,q)-graph G =( V,E) is said to be magic if there exists a bijection f : V ∪ E →{ 1, 2, 3,...,p+ q} such that for all edges uv of G, f (u )+ f (v )+ f (uv) is a constant. The minimum of all constants say, m(G), where the minimum is taken over all such bijections of a magic graph G, is called the magic strength of G. In this paper we define the maximum of all constants say, M (G), analogous to m(G), and introduce strong magic, ideal magic, weak magic labelings, and prove that some known classes of graphs admit such labelings.