On edge irregularity strength of graphs

For a simple graph G  , a vertex labeling ϕ:V(G)→{1,2,…,k}ϕ:V(G)→{1,2,…,k} is called k-labeling. The weight of an edge xy in G  , denoted by wϕ(xy)wϕ(xy), is the sum of the labels of end vertices x and y  , i.e. wϕ(xy)=ϕ(x)+ϕ(y)wϕ(xy)=ϕ(x)+ϕ(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 wϕ(e)≠wϕ(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)es(G).In this paper, we estimate the bounds of the edge irregularity strength and determine the exact value for several families of graphs.