Abstract
In this paper, we introduce super weak sum labeling and weak sum labeling of a graph G with vertex set V and edge set E, defined as follows: A super weak sum (briefly sw-sum) labeling is a bijection L : V -> {1, 2,...,vertical bar V vertical bar} such that for every edge (u, v) in G, there is a vertex w in G with L(u) + L(v) = L(w). A graph that can be sw-sum labeled is called an sw-sum graph. It is obvious that an sw-sum graph cannot be connected. There must be at least one isolated vertex, namely the vertex with the largest label. The sw-sum number, w(H), of a connected graph H is the least number r of isolated vertices (K-r) over bar such that G = H boolean OR (K-r) over bar, is an sw-sum graph. If the set {1, 2,...,vertical bar V vertical bar} is replaced by some subset S of Z(+) in the definition of sw-sum labeling, then such a labeling will be referred to as weak sum (briefly w-sum) labeling and the minimum number of isolates in such a labeling as w-sum number. We show that a lower bound for the sw-sum number is the minimum degree delta of a vertex in the graph. Graphs achieving this bound will be referred to as delta-optimal sw-summable. We provide labeling schemes for different families of graphs showing that they are delta-optimal sw-summable. We show that not all the graphs are delta-optimal sw-summable and conjecture that all the graphs are delta-optimal w-summable.