On the reflexive edge strength in zigzag graphs

Articles in Press

Document Type : Research Paper

Authors

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan

2 Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia

3 School of Computing and Data Science, Xiamen University Malaysia, 43900 Sepang, Selangor, Malaysia

4 Mathematics Department, University of Jember, Jl. Kalimantan N0. 37 Sumbersari Jember, East Java 68121, Indonesia

Abstract

Let $G$ be a simpple graph. The total $k$-labeling of $G$ is the assignments of a non-negative integer from the set $\{0,2,\ldots,2\lfloor k/2\rfloor\}$ to the vertices and a positive integer from the set $\{1,2,\ldots,k\}$ to the edges of a graph $G$. It is called an edge irregular reflexive $k$-labeling of the graph $G$ if the weights of any two different edges are distinct, where the edge weight is the sum of the label of the edge itself and the labels of its two end vertices. The minimum value $k$ for which the graph $G$ has an edge irregular reflexive $k$-labeling is called the reflexive edge strength of $G$. In this paper, we determine the exact value of the reflexive edge strength for the zigzag graph $Z_{m}^{n}$, where $n \geq 2 $ and $m \geq 3$.

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References

[1] I. H. Agustin, Dafik, M. I. Utoyo, Slamin and M. Venkatachalam, The reflexive edge strength on some almost regular graphs, Heliyon, 7 no. 5 (2021) e06991.
[2] A. Ahmad, M. K. Siddiqui and D. Afzal, On the total edge irregularity strength of zigzag graphs, Australas. J. Combin., 54 (2012) 141–149.
[3] M. A. Asif, R. Ismail, A. Razaq, E. H. A. Al-Sabri, M. H. Mateen and S. Ali, An application on edge irregular reflexive labeling for mt-graph of cycle graph, AIMS Math., 10 no. 1 (2025) 1300–1321.
[4] M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ıkov´a, and D. Tanna, On edge irregular reflexive labelings for the generalized friendship graphs, Mathematics, 67 (2017) 2–11.
[5] M. Baˇca, M. Irfan, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ıkov´a and D. Tanna, Note on edge irregular reflexive labelings of graphs, AKCE Int. J. Graphs Comb., 16 no. 2 (2019) 145–157.
[6] M. Baˇca, S. Jendrol’, M. Miller, and J. Ryan, On irregular total labelings, Discrete Math., 307 (2007) 1378–1388.
[7] M. Basher, On the reflexive edge strength of the circulant graphs, AIMS Math., 6 no. 9 (2021) 9342–9365.
[8] M. Basher, The reflexive edge strength of toroidal fullerene, AKCE Int. J. Graphs Comb., 20 no. 1 (2023) 35–39.
[9] G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer., 64 (1988) 187–192.
[10] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Comb., 5 2023, #DS 6.
[11] J. L. G. Guirao, S. Ahmad, M. K. Siddiqui and M. Ibrahim, Edge irregular reflexive labeling for the disjoint union of generalized Petersen graph, Mathematics, 6 (2018), 304.
[12] M. Irfan, M. Baˇca and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a, On reflexive edge strength of generalized prism graphs, Electron. J. Graph Theory Appl., 10 no. 2 (2022) 415–424.
[13] Y. Ke, M. J. A. Khan, M. Ibrahim and M. K. Siddiqui, On edge irregular reflexive labeling for cartesian product of two graphs, Eur. Phys. J. Plus, 136 article number 6 (2021).
[14] M. J. A. Khan, M. Ibrahim and A. Ahmad, On edge irregular reflexive labeling of categorical product of two paths, Comput. Syst. Sci. Eng., 36 no. 3 (2021) 485–492.
[15] J. Ryan, B. Munasinghe, A. Semaniˇcov´a-Feˇnovˇc´ıkov´a and D. Tanna, Reflexive irregular labelings, submitted.
[16] D. Tanna, J. Ryan and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a, Edge irregular reflexive labeling of prisms and wheels, Australas. J. Combin., 69 (2017) 394–401.
[17] 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 no. 1 (2021) 53–59.
[18] K. K. Yoong, R. Hasni, G. C. Lau, M. A. Asim and A. Ahmad, Reflexive edge strength of convex polytopes and corona product of cycle with path, AIMS Math., 7 no. 7 (2022) 11784–11800.
[19] K. K. Yoong, R. Hasni, G. C. Lau, and M. Irfan, Edge irregular reflexive labeling for some classes of plane graphs, Malays. J. Math. Sci., 16 no. 1 (2022) 25–36.
[20] X. Zhang, M. Ibrahim, S. A. H. Bokhary and M. K. Siddiqui, Edge irregular reflexive labeling for the disjoint union of gear graphs and prism graphs, Mathematics, 6 no. 9 (2018) 142.
Transactions on Combinatorics

Articles in Press, Corrected Proof
Available Online from 26 May 2026

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History

  • Receive Date: 01 September 2025
  • Revise Date: 06 October 2025
  • Accept Date: 25 January 2026
  • Published Online: 26 May 2026

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