1-Distant Irregularity Strength of Graphs

Abstract

Let $G=(V,E)$ be a connected graph of order $n\ge 3$. Let $f:E\rightarrow \{1, 2,...,k\}$ be a function and let the weight of a vertex v be defined by $\omega (v)= \sum \limits _{v \in V} f(v)$. Then f is called an irregular labeling if all the vertex weights are distinct. The irregularity strength s(G) is the smallest positive integer k such that there is an irregular labeling $f:E\rightarrow \{1, 2,...,k\}$. In this paper we prove that for some families of graphs, irregularity strength and r-distant irregularity strength are equal. Further exact value of 1-distant irregularity strength of some classes of graphs are determined.