On Total Irregularity Strength of Double-Star and Related Graphs

Abstract

Abstract Let G = ( V , E ) be a simple and undirected graph with a vertex set V and an edge set E . A totally irregular total k -labeling f : V ∪ E → {1, 2,. . ., k } is a labeling of vertices and edges of G in such a way that for any two different vertices x and x 1, their weights and are distinct, and for any two different edges xy and x 1 y 1 their weights f ( x ) + f ( xy ) + f ( y ) and f ( x 1) + f ( x 1 y 1) + f ( y 1) are also distinct. A total irregularity strength of graph G , denoted byts( G ), is defined as the minimum k for which G has a totally irregular total k -labeling. In this paper, we determine the exact value of the total irregularity strength for double-star S n , m , n , m ≥ 3 and graph related to it, that is a caterpillar S n ,2, n , n ≥ 3. The results are and ts ( S n ,2, n ) = n .