In this paper we introduce a new labeling of graphs called d -edge sum labeling and study the same as a vertex coloring problem. Let l: E(G)→N be a labeling of the edges of a graph G by positive integers. Define c(u)=∑(u,v)∈E(G) l(u,v)+d(u) where d(u) denotes the degree of u . We call l a d -edge sum labeling if c(u) ≠ c(v), for every pair of adjacent vertices u and v in G . The d -edge sum number of a graph G, denoted by ηds (G), is the least positive k such that G has a d -edge sum labeling with (1,2,...,k) as the set of labels. In this paper we obtain η ds(G) for some special classes of graphs.