### On inclusive distance vertex irregularity strength of book graph

#### Abstract

The concept of distance vertex irregular labeling of graphs was introduced by Slamin in 2017. The *distance vertex irregular labeling* on a graph *G* with *v* vertices is defined as an assignment *λ* : V → {1, 2, ..., *k*} so that the weights calculated at vertices are distinct. The weight of a vertex *x* in *G* is defined as the sum of the labels of all the vertices adjacent to *x* (distance 1 from *x*). The distance vertex irregularity strength of graph *G*, denoted by *dis(G)*, is defined as the minimum value of the largest label *k* over all such irregular assignments. Bong, Lin and Slamin generalized the concept to inclusive and non-inclusive distance irregular labeling. The difference between them depends on the way to calculate the weight on the vertex whether the vertex label we calculate its weight is included or not. The *inclusive distance vertex irregularity strength* of *G*, is defined as the minimum of the largest label *k* over all such inclusive irregular assignments. In this paper, we determine the inclusive distance vertex irregularity strength of some particular classes of graphs such as book graph.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.19184/ijc.2023.7.2.4

#### References

S. Arumugam, D. Forencek and N. Kamatchi, Distance magic graphs-survey, *J. Indones. Math. Soc.*, Special Ed (2011), 11-26.

Baca. M, A.S-Fenovcikova, Slamin, and K.A. Sugeng, On inclusive distance vertex irreguler labelings, *Electron. J. Graph Theory Appl.*, **6**(1) (2018), 61-83.

F Susanto, C N Betistiyan, I Halikin, and K wijaya. On inclusive distance vertex irregularity strength of small identical copies of star graphs, *J. Phys (2021) : Conf Ser.* 1872 012005

N.H. Bong, Y. Lin and Slamin, On inclusive and non-inclusive vertex irregular d-distance vertex labelings, JCMCC, in press.

G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular network, *Congr. Numer.*, **64**(1998), 187-192.

M. Miller, C. Rodger and R. Simanjuntak, Distance magic labelings of graphs, *Australas. J. Combin.*, **28**(2003), 305-315.

Slamin, On distance irregular labelings of graphs, *Far East J. Math. Sci. (FJMS)*, **102**(5), (2017), 919-932.

U. Budi, K.A. Sugeng and S. Utama, On inclusive distance vertex irregular 1-distance labelings on triangular ladder graphs, *AIP Conf. Proc. 2021*, (2018), 060006-1-060006-6.

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.