On the total vertex irregularity strength of trees

A vertex irregular total kk-labelling λ:V(G)∪E(G)⟶{1,2,…,k}λ:V(G)∪E(G)⟶{1,2,…,k} of a graph GG is a labelling of vertices and edges of GG done in such a way that for any different vertices xx and yy, their weights wt(x)wt(x) and wt(y)wt(y) are distinct. The weight wt(x)wt(x) of a vertex xx is the sum of the label of xx and the labels of all edges incident with xx. The minimum kk for which a graph GG has a vertex irregular total kk-labelling is called the total vertex irregularity strength of GG, denoted by tvs(G). In this paper, we determine the total vertex irregularity strength of trees.