Total vertex irregularity strength of trees with maximum degree four

Abstract

Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a vertex x is defined as wt(x) = ϕ(x) + ∑y∈N(x) ϕ (xy), where N(x) is the set of neighbors of x and y. The labeling ϕ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y we have wt(x) ≠ wt(y). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper, we determine total vertex irregularity strengths of trees with maximum degree four and a subdivision of a double-star.