Elegant Labeling Of Some Graphs

Abstract

In this paper, we introduce a new notion of graph theory study, namely a local edge metric dimension. It is a natural extension of metric dimension concept. dG(e, v) = min{d(x, v), d(y, v)} is the distance between the vertex v and the edge xy in graph G. A non empty set S ⊂ V is an edge metric generator for G if for any two edges e1, e2 ∈ E there is a vertex k ∈ S such that dG(k, e1 6= dG(k, e2)). The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dimE(G). The local edge metric dimension of G, denoted by dimlE(G), is a local edge metric generator of G if r(xk|S) 6= r(yk|S) for every pair xk, ky of adjacent edges of G. Our concern in this paper is investigating some results of local edge metric dimension on some graphs