On total vertex irregularity strength of hexagonal cluster graphs (Q2033816)

Summary: For a simple graph \(G\) with a vertex set \(V\) and an edge set \(E\), a labeling \(f:V \bigcup E\longrightarrow \{1,2,\cdots,k\}\) is called a vertex irregular total \(k\)-\text{labeling} of \(G\) if for any two different vertices \(x\) and \(y\) in \(V\) we have \(wt (x)\neq wt (y)\) where \(wt (x)=f(x)+\sum_{u \in V} f(xu)\). The smallest positive integer \(k\) such that \(G\) has a vertex irregular total \(k\)-\text{labeling} is called the total vertex irregularity strength of \(G\), denoted by \(tvs\). The lower bound of \(tvs\) for any graph \(G\) have been found by \textit{M. Bača} et al. [Discrete Math. 307, No. 11--12, 1378--1388 (2007; Zbl 1115.05079)]. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on \(n\) cluster for \(n\geq 2\). Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on \(n\) cluster is \((3n^2 +1)/2\).