On Vertex Irregular Total Labelings

Abstract

A \({vertex \;irregular\; total \;labeling}\) \(\sigma\) of a graph \(G\) is a labeling of vertices and edges of \(G\) with labels from the set \(\{1, 2, \ldots, k\}\) in such a way that for any two different vertices \(x\) and \(y\), their weights \(wt(x)\) and \(wt(y)\) are distinct. The \({weight}\) \(wt(x)\) of a vertex \(x\) in \(G\) is the sum of its label and the labels of all edges incident with \(x\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total labeling is called the \({total \;vertex\; irregularity \;strength}\) of \(G\). In this paper, we study the total vertex irregularity strength for two families of graphs, namely Jahangir graphs and circulant graphs.