Some new formulae on Δ–optimum exclusive sum labeling of certain trees. (3rd March 2016) - Samal, R. K.; Mishra, D.
Abstract: All sum graphs are disconnected. The notions of sum labeling and sum graphs were introduced by Harary [1] and later extended to include all integers [2]. A mapping L is called a sum labeling a graph H ( V ( H ), E ( H )) if it is an injection from V ( H ) to a set of positive integers such that xy ∈ E ( H ) if and only if there exists a vertex w ∈ V ( H ) such that L ( w )= L ( x ) + L ( y ). In this case w is called a working vertex. We call L as an exclusive sum labeling of a graph G if it is sum labeling of for some non negative integer r and G contains no working vertex. In general, a graph G will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called exclusive sum number of G, denoted by ∈( G ). An exclusive sum labeling of a graph G is said to be optimum if it labels G exclusively by using ∈( G ) isolated vertices. In case ∈( G )=Δ( G ), where D( G ) denotes the maximum degree of vertices in G, the labeling is called Δ- optimum exclusive sum labeling. In this paper we developed some new formulae to find Δ- optimum exclusive sum labeling of certain trees.