On the Total Vertex Irregularity Strength of Series Parallel Graph sp(m,r,3)

Authors

  • Corry Corazon Marzuki Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru, Indonesia
  • Muzdhalifah Marinka Utami Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru, Indonesia
  • Fitri Aryani Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru, Indonesia
  • Mona Elviyenti Politechnic Caltex Riau, Pekanbaru, Indonesia
  • Ade Novia Rahma Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru, Indonesia

DOI:

https://doi.org/10.30983/lattice.v5i2.10483

Keywords:

Series parallel graph, Total vertex irregularity strength, Total vertex irregular labelling

Abstract

This paper addresses the problem of determining the total vertex irregularity strength of the series-parallel graph family  for  and. The total vertex irregularity strength  of a graph  is defined as the smallest integer  such that there exists a total k-labelling  where the vertex weights  are distinct for each vertex. The graph family  is generated through repeated series and parallel compositions, with parameters and a fixed structural parameter 3. To solve this problem, we construct an explicit total labelling that ensures distinct vertex weights, providing an upper bound for . Additionally, we perform a structural analysis of the graph, which yields a matching lower bound. The results demonstrate that the total vertex irregularity strength of  is given by . This work contributes a new insight into the characterization of the total vertex irregularity strength for this specific class of graphs, providing both upper and lower bounds for .

 

Penelitian ini membahas permasalahan dalam menentukan nilai total ketakteraturan titik pada keluarga graf seri-paralel  untuk  dan . Nilai total ketakteraturan titik  untuk suatu graf  didefinisikan sebagai nilai minimum  sehingga terdapat pelabelan total  dengan bobot titik  yang berbeda untuk setiap titik. Keluarga graf  dibangun melalui komposisi seri dan paralel secara berulang, dengan parameter  dan parameter struktur tetap 3. Untuk menyelesaikan masalah ini, kami mengonstruksi pelabelan total eksplisit yang memastikan bobot titik saling berbeda, sehingga menghasilkan batas atas untuk . Selain itu, kami melakukan analisis struktur graf untuk memperoleh batas bawah yang sesuai. Hasil penelitian ini menunjukkan bahwa total vertex irregularity strength untuk  diberikan oleh . Penelitian ini memberikan kontribusi berupa wawasan baru dalam karakterisasi nilai total ketakteraturan titik untuk kelas graf ini, dengan menyediakan batas atas dan batas bawah untuk

Author Biographies

Muzdhalifah Marinka Utami, Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru

Department of Mathematics

Fitri Aryani, Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru

Department of Mathematics

Mona Elviyenti, Politechnic Caltex Riau, Pekanbaru

Electronical Telecomunication Department

Ade Novia Rahma, Universitas Islam Negeri Sultan Syarif Kasim Riau, Pekanbaru

Department of Mathematics

References

E. Kay, J. A. Bondy, and U. S. R. Murty, “Graph Theory with Applications,” Oper. Res. Q., vol. 28, no. 1, p. 237, 1977, doi: 10.2307/3008805.

R. W. Putra and Y. Susanti, “On total edge irregularity strength of centralized uniform theta graphs,” AKCE Int. J. Graphs Comb., vol. 15, no. 1, pp. 7–13, 2018, doi: 10.1016/j.akcej.2018.02.002.

J. A. Gallian, “A Dynamic Survey of Graph Labeling,” Electron. J. Comb., 2018.

R. Munir, “Matematika Disktrit,” Inform. Bandung, pp. 281–308, 2010.

G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz, and F. Saba, “Irregular Networks,” Congr. Numer., vol. 64, pp. 197–210, 1988.

Z. B. Tomi, M. F. Akbar, F. F. Janeva, G. H. Medika, and K. Kunci, “Pelabelan Vertex Graceful pada Graf- ( 5 , 8 ),” vol. 3, no. 1, pp. 63–76, 2025.

G. H. Medika and others, “Pelabelan Vertex-Graceful pada Graf-(5,7),” Lattice J. Math., 2024.

G. H. Medika, A. Budiman, and R. Yolanda, “Pelabelan Graceful Titik pada Graf-(7,8),” J. Math. Struct., 2025.

M. Bača, S. Jendrol’, M. Miller, and J. Ryan, “On irregular total labellings,” Discrete Math., vol. 307, no. 11–12, pp. 1378–1388, 2007, doi: 10.1016/j.disc.2005.11.075.

N. Hinding, D. Firmayasari, H. Basir, M. Baˇ, and A. Semaniˇ, “On irregularity strength of diamond network,” AKCE Int. J. Graphs Comb., vol. 15, pp. 291–297, 2018, doi: 10.1016/j.akcej.2017.10.003.

T. Winarsih and D. Indriati, “Total edge irregularity strength of (n,t)-kite graph,” J. Phys. Conf. Ser., vol. 1008, no. 1, pp. 449–456, 2018, doi: 10.1088/1742-6596/1008/1/012049.

R. Ramdani, “On the total vertex irregularity strength of comb product of two cycles and two stars,” Indones. J. Comb., vol. 3, no. 2, pp. 79–94, 2019, doi: 10.19184/ijc.2019.3.2.2.

C. C. Marzuki, “Nilai Total Ketakteraturan Sisi dari m-copy Graf Lintasan,” J. Sains Mat. dan Stat., vol. 5, no. 1, pp. 90–98, 2019.

I. Rosyida and D. Indriati, “Determining Total Vertex Irregularity Strength of Tr(4,1) Tadpole Chain Graph and its Computation,” Procedia Comput. Sci., vol. 157, pp. 699–706, 2019, doi: 10.1016/j.procs.2019.09.152.

Y. Susanti, Y. Indah, and H. Khotimah, “On Total Edge Irregularity Strength of Staircase Graphs and Related Graphs,” vol. 15, no. 1, pp. 1–13, 2020, doi: 10.21859/IJMSI.15.1.1.

N. Hinding, H. K. Kim, N. Sunusi, and R. Mise, “On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs,” Int. J. Math. Math. Sci., vol. 2021, 2021, doi: 10.1155/2021/2743858.

S. T. Rajasingh, Indra, Arockiamary, “Total Edge Irregularity Strength of Series Parallel Graphs,” Int. J. Pure Appl. Math., vol. 99, no. 1, pp. 11–21, 2015.

C. C. Marzuki, L. Laraza, and F. Aryani, “NILAI TOTAL KETAKTERATURAN TITIK PADA GRAF SERI PARALEL sp(m,1,3),” J. Sains Mat. dan Stat., vol. 6, no. 2, p. 113, 2020, doi: 10.24014/jsms.v6i2.10559.

H. Riskawati, Nurdin, “Nilai ketidakteraturan pada graf series parallel,” vol. 1, no. 1, pp. 5–10, 2019.

Downloads

Published

2025-12-30

How to Cite

Marzuki, C. C., Utami, M. M., Aryani, F., Elviyenti, M., & Rahma, A. N. (2025). On the Total Vertex Irregularity Strength of Series Parallel Graph sp(m,r,3). Lattice Journal : Journal of Mathematics Education and Applied, 5(2), 160–175. https://doi.org/10.30983/lattice.v5i2.10483

Issue

Vol. 5 No. 2 (2025): Desember 2025

Section

Articles

Citation Check

License

Copyright (c) 2025 Corry Corazon Marzuki, Muzdhalifah Marinka Utami, Fitri Aryani, Mona Elviyenti, Ade Novia Rahma

Creative Commons License

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

Authors who publish with Lattice Journal : Journal of Mathematics Education and Applied agree to the following terms: Authors retain copyright and grant the Lattice Journal : Journal of Mathematics Education and Applied right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY-SA 4.0) that allows others to share (copy and redistribute the material in any medium or format) and adapt (remix, transform, and build upon the material) the work for any purpose, even commercially with an acknowledgement of the work's authorship and initial publication in Lattice Journal : Journal of Mathematics Education and Applied. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in Lattice Journal : Journal of Mathematics Education and Applied. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).