Sum number of fans

Graph labeling deals with assigning labels to one or more elements of a graph. It has a wide variety of applications including: coding theory, communication network addressing, data base management system and secret sharing schemes to mention a view. A mapping [Formula: see text] is called a sum labeling of a graph [Formula: see text] if it is an injection from [Formula: see text] to a set of positive integers, such that [Formula: see text] if and only if there exists a vertex [Formula: see text] such that [Formula: see text]. In this case, [Formula: see text] is called a working vertex. In general, a graph [Formula: see text] will require some isolated vertices to be labeled in this way. The least possible number of such isolated vertices is called the sum number of [Formula: see text]; denoted by [Formula: see text]. A sum labeling of a graph [Formula: see text] is said to be optimum if it labels [Formula: see text] by using [Formula: see text] isolated vertices. In this paper, we investigate the lower bounds for the number of isolates required by an even fan and an odd fan, and then we construct optimum sum labelling for the graphs to prove: [Formula: see text]