On Vertex Irregular Total k-labeling and Total Vertex Irregularity Strength of Lollipop Graphs

Abstract

Abstract Let G be a connected graph with vertex set V(G) and edge set E(G). A vertex irregular total k-labeling λ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } of a graph G is a labeling of vertices and edges of G in such a way that the weights of any two different vertices x and y are distinct. The weight of a vertex x in G, denoted by wt(x), is defined as the sum of the label of x and the labels of all edges incident with the vertex x. The total vertex irregularity strength of G, denoted by tvs(G), is the smallest positive integer k for which the graph G has a vertex irregular total k-labeling. The (m, n)-lollipop graphs denoted by Lm,n is a graph obtained by joining a complete graph Km to a path graph Pn with a bridge. In this research, we investigate tvs of lollipop graphs Lm,n for m ≥ 3 and n ≥ 1, denoted by tvs(Lm,n ).