Upper bounds on distance vertex irregularity strength of some families of graphs

Abstract

For a graph $G$ its distance vertex irregularity strength is the smallest integer $k$ for which one can find a labeling $f: V(G)\to {1, 2, \dots, k}$ such that $\sum_{x\in N(v)}f(x)\neq \sum_{x\in N(u)}f(x)$ for all vertices $u,v$ of $G$, where $N(v)$ is the open neighborhood of v. In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees.