On the vertex irregular reflexive labeling of generalized friendship graph and corona product of graphs

Document Type : Original paper

Authors

1 Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Terengganu, Malaysia

2 Universiti Teknologi MARA, Faculty of Computer and Mathematical Sciences, 85100 Segamat, Johor, Malaysia

3 College of Computer Sciences and Information Technology, Jazan University, Jazan, Saudi Arabia

Abstract

For a graph $G$, we define a total $k$-labeling $\varphi$ as a combination of an edge labeling $\varphi_e:E(G)\rightarrow \{1,\,2,\,\ldots,\,k_e\}$ and a vertex labeling $\varphi_v:V(G)\rightarrow \{0,\,2,\,\ldots,\,2k_v\}$, where $k=\,\mbox{max}\, \{k_e,2k_v\}$. The total $k$-labeling $\varphi$ is called a vertex irregular reflexive $k$-labeling of $G$ if any pair of vertices $u$, $u'$ have distinct vertex weights $wt_{\varphi}(u)\neq wt_{\varphi}(u')$, where $wt_{\varphi}(u)=\varphi(u)+\sum_{uu'\in E(G)} \varphi(uu')$ for any vertex $u\in V(G)$. The smallest value of $k$ for which such a labeling exists is called the reflexive vertex strength of $G$, denoted by $rvs{(G)}$. In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration.

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References

[1] I.H. Agustin, L. Susilowati, Dafik, I.N. Cangul, and N. Mohanapriya, On the vertex irregular reflexive labeling of several regular and regular-like graphs, J. Discrete Math. Sci. Cryptogr 25 (2022), no. 5, 1457–1473.
https://doi.org/10.1080/09720529.2022.2063543
[2] I.H. Agustin, M.I. Utoyo, Dafik, and M. Venkatachalam, The vertex irregular reflexive labeling of some almost regular graph, Palest. J. Math. 10 (2021), no. Special Issue II, 83–91.
[3] I.H. Agustin, M.I. Utoyo, and M. Venkatachalam, On the construction of the reflexive vertex k-labeling of any graph with pendant vertex, Int. J. Math. Math. Sci. 2020 (2020), Article ID: 7812812.
https://doi.org/10.1155/2020/7812812
[4] R. Alfarisi, J. Ryan, M.K. Siddiqui, Dafik, and I.H. Agustin, Vertex irregular reflexive labeling of disjoint union of gear and book graphs, Asian-Eur. J. Math. 14 (2021), no. 5, Article ID: 2150078.
https://doi.org/10.1142/S1793557121500789
[5] M. Bača, S. Jendrol', M. Miller, and J. Ryan, On irregular total labellings, Discrete Math. 307 (2007), no. 11-12, 1378–1388.
https://doi.org/10.1016/j.disc.2005.11.075
[6] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, and F. Saba, Irregular networks, Congr. Numer. 64 (1988), 197–210.
[7] G. Chartrand and P. Zhang, A First Course in Graph Theory, Dover Publications, Inc., Mineola, New Tork, 2013.
[8] H. Fernau, J.F. Ryan, and K.A. Sugeng, A sum labelling for the generalised friendship graph, Discrete Math. 308 (2008), no. 5-6, 734–740.
https://doi.org/10.1016/j.disc.2007.07.059
[9] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Comb. (2022), no. DynamicSurveys, Article ID: DS6.
[10] J. Ryan, B. Munasinghe, A. Semaničová-Feňovčíková, and D. Tanna, Reflexive irregular labelings, submitted.
[11] Z. Ryjáček and I. Schiermeyer, The flower conjecture in special classes of graphs, Discuss Math. Graph Theory 15 (1995), no. 2, 179–184.
[12] M.S. Saleem, M. Irfan, and M.S. Rao, Reflexive vertex strength of generalized petersen graph, submitted.
[13] D. Tanna, J. Ryan, A. Semaničová-Feňovčíková, and M. Bača, Vertex irregular reflexive labeling of prisms and wheels, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 51–59.
https://doi.org/10.1016/j.akcej.2018.08.004
[14] K.K. Yoong, R. Hasni, M. Irfan, I. Taraweh, A. Ahmad, and S.M. Lee, On the edge irregular reflexive labeling of corona product of graphs with path, AKCE Int. J. Graphs Comb. 18 (2021), no. 1, 53–59.
https://doi.org/10.1080/09728600.2021.1931555
Communications in Combinatorics and Optimization
Volume 9, Issue 3
September 2024
Pages 509-526

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