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 λ is called a sum labeling of a graph H (V (H) , E (H)) if it is an injection from V (H) to a set of positive integers, such that { x , y } ∈ E (H) if and only if there exists a vertex w ∈ V (H) such that λ (w) = λ (x) + λ (y). In this case, w is called a working vertex. In general, a graph G 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 G ; denoted by σ (G). A sum labeling of a graph G is said to be optimum if it labels G by using σ (G) 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: σ (F n) = 3 for even  n , n ≥ 4 , and for  n = 3 4 for odd  n , n ≥ 5. [ABSTRACT FROM AUTHOR]