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

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 B n is obtained by connecting two copies of a complete graph K n by a bridge. In this research, we determined the total edge irregularity strength of barbell graph B n for n ≥ 3.