On the total vertex irregularity strength of trees

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