On Vertex Irregular Total Labelings

Abstract

A $vertexirregulartotallabeling$ $\sigma$ of a graph $G$ is a labeling of vertices and edges of $G$ with labels from the set $\{1,2,\dots ,k\}$ 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 $totalvertexirregularitystrength$ of $G$. In this paper, we study the total vertex irregularity strength for two families of graphs, namely Jahangir graphs and circulant graphs.