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.

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