On edge irregularity strength of graphs

For a simple graph G, a vertex labeling @f:V(G)->{1,2,...,k} is called k-labeling. The weight of an edge xy in G, denoted by w"@f(xy), is the sum of the labels of end vertices x and y, i.e. w"@f(xy)=@f(x)+@f(y). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f there is w"@f(e) w"@f(f). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we estimate the bounds of the edge irregularity strength and determine the exact value for several families of graphs.