On the total vertex irregularity strength of firecracker graph

Abstract

Let G be a connected graph with vertex set V (G) and edge set E(G). A total k-labeling is a labeling . A vertex irregular total k-labeling of a graph G(V; E) is a total k-labeling such that the weight for all vertices is distinct. The weight of vertex u in G, denoted by wt(u), is defined as the sum of the label of vertex u and the label of all edges incident with the vertex u. Total vertex irregularity strength of G denoted by tvs(G), is the minimum value of the largest label k over all such vertex irregular total k-labeling. The firecracker graph is a graph obtained by the concatenation of mn-stars by linking one leaf from each. In this paper, we investigate the total vertex irregularity strength of firecracker graphs, for n = 1, 2 and m > 1. For n = 1 and m > 1. For n = 2 and m > 1, is defined into 2 formulas; for 2 ≤ m ≤ 8 and for m ≥ 9.