On Edge Irregular Total k-labeling and Total Edge Irregularity Strength of Barbell Graphs

Abstract

Let G be a connected graph with a non empty vertex set V(G) and edge set E(G). An edge irregular total k-labeling of a graph G is a labeling λ : V(G) ⋃ E(G) → {1, 2, …, k}, so that every two different edges have different weights. The weight of edge uv of G is the sum of the labels vertices u and v and label of the edge uv, which is can be written as wt(uv) = λ (u) + λ (uv) + λ(v). The total edge irregularity strength of G, denoted by tes(G) is the minimum positive integer k for which the graph G has an edge irregular total k-labeling. Barbell graph Bn is obtained by connecting two copies of a complete graph Kn by a bridge. In this research, we determined the total edge irregularity strength of barbell graph Bn for n ≥ 3.