On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

For a simple graph G with a vertex set VG and an edge set EG, a labeling f:VG∪​EG⟶1,2,⋯,k is called a vertex irregular total k−labeling of G if for any two different vertices x and y in VG we have wtx≠wty where wtx=fx+∑u∈VGfxu. The smallest positive integer k such that G has a vertex irregular total k−labeling is called the total vertex irregularity strength of G, denoted by tvsG. The lower bound of tvsG for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n≥2. Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3n2+1/2.