All graph in this paper is connected and simple graph. Let d(u, v) be a distance between any vertex u and v in graph G = (V, E ). A function : ( ) l V G {1, 2, , } k →  is called vertex irregular k-labelling ...

A function f is called a local irregularity vertex coloring if (i) l : V (G) ! f1; 2; ; kg as vertex irregular k-labeling and w : V (G) ! N, for every uv 2 E(G); w(u) 6= w(v) where w(u) = v2N(u)l(v) and (ii) ...