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

AbstractWe 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 wtG(v)=∑e∋vw(e)+w(v) for each v∈V. The smallest s for which exists such a weighting with wtG(u)≠wtG(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)≤3⌈n/δ⌉+1 for each graph of order n and with minimum degree δ>0