A new upper bound for the total vertex irregularity strength of graphs

We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting w of a graph G=(V,E) with elements of a set {1,2,...,s}, denote wt"G(v)[email protected]?"e"@?"vw(e)+w(v) for each [email protected]?V. The smallest s for which exists such a weighting with wt"G(u) wt"G(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by tvs(G). We prove that tvs(G)@[email protected]?n/@[email protected]?+1 for each graph of order n and with minimum degree @d>0.