Edge Metric Dimension of Uniform Theta Graphs and Sunflower Graphs

Muchammad Abrori; Deddy Rahmadi; Lailatul Ulla Safitri; Alvian Nur Rohman; Jaqueline Widad Zuha

doi:10.20956/t02sds33

Authors

Muchammad Abrori UIN Sunan Kalijaga Yogyakarta
Deddy Rahmadi Universitas Islam Negeri Sunan Kalijaga
Lailatul Ulla Safitri UIN Sunan Kalijaga Yogyakarta
Alvian Nur Rohman UIN Sunan Kalijaga Yogyakarta
Jaqueline Widad Zuha UIN Sunan Kalijaga Yogyakarta

DOI:

https://doi.org/10.20956/t02sds33

Keywords:

edge metric dimension, edge resolving set, uniform theta graph, sunflower graph

Abstract

Let be a graph with vertex set and edge set . In graph theory, an edge resolving set is defined as a subset of vertices such that every edge in has a distinct distance representation with respect to . The minimum cardinality of an edge resolving set is called the edge metric dimension of , denoted by . This study aimed to determine the edge metric dimension of the uniform theta graph and the sunflower graph . The method employed a graph-structure approach and distance analysis between edges and vertices supported by a literature review. The results showed that for and , for , and for . Furthermore, for and , for and , for , and for .

References

[1] Abbas, S., Raza, Z., Siddiqui, N., Khan, F., & Whangbo, T. (2022). Edge metric dimension of honeycomb and hexagonal networks for IoT. Computers, Materials and Continua, 71, 2683–2695.

[2] Abbas, W., Chaudhry, F., Farooq, U., Azeem, M., & Shang, Y. (2024). Investigating metric dimension and edge metric dimension of hexagonal boron nitride and carbon nanotubes. European Journal of Pure and Applied Mathematics, 17, 2055–2072.

[3] Adawiyah, R., Dafik, Alfarisi, R., Prihandini, R. M., & Agustin, I. H. (2018). Edge metric dimension on some families of tree. Journal of Physics: Conference Series. [Online]. Available: https://iopscience.iop.org/article/10.1088/1742-6596/1180/1/012005

[4] Ahsan, M., Zahid, Z., & Zafar, S. (2020). Edge metric dimension of some classes of circulant graphs. Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 28, 15–37.

[5] Buhaerah, Z., Busrah, & Sanjaya, H. (2022). Teori graf dan aplikasinya. [Online]. Available: https://repository.iainpare.ac.id/id/eprint/3489/

[6] Chandran, S. C., & Reji, T. (2024). Edge metric dimension of some Cartesian product of graphs. Proyecciones, 43, 587–611.

[7] Chartrand, G., Salehi, E., & Zhang, P. (2000). The partition dimension of resolvability in graphs. Aequationes Mathematicae, 59, 45–54.

[8] Farooq, M. U., et al. (2024). Computing edge version of metric dimension of certain chemical networks. Scientific Reports, 14. https://doi.org/10.1038/s41598-024-62063-6

[9] Filipović, V., Kartelj, A., & Kratica, J. (2019). Edge metric dimension of some generalized Petersen graphs. Results in Mathematics, 74. https://doi.org/10.1007/s00025-019-1105-9

[10] Kelenc, A., Tratnik, N., & Yero, I. G. (2018). Uniquely identifying the edges of a graph: The edge metric dimension. Discrete Applied Mathematics. https://doi.org/10.1016/j.dam.2018.05.052

[11] Knor, M., Majstorović, S., Masa Toshi, A. T., Škrekovski, R., & Yero, I. G. (2021). Graphs with the edge metric dimension smaller than the metric dimension. Applied Mathematics and Computation, 401. https://doi.org/10.1016/j.amc.2021.126076

[12] Kusmayadi, T. A., & Rahmadi, D. (2025). Dimensi metrik dari graf. Star Digital Publishing.

[13] Mhagama, L. S., Husin, M. N., Nadeem, M. F., & Ali, W. (2025). Computing the edge metric dimension of the line graph of molecular graphs of some classes of antiviral drugs. Contemporary Mathematics (Singapore), 6, 1784–1802.

[14] Mhagama, L. S., Nadeem, M. F., & Husin, M. N. (2024). On the edge metric dimension of some classes of cacti. AIMS Mathematics, 9, 16422–16435.

[15] Mn, A. N. A., Hasmawati, & Nurwahyu, B. (2025). The edge metric dimension of comb product cycles versus stars and fans. Journal of Advanced Research in Applied Sciences and Engineering Technology, 49, 52–63.

[16] Peterin, I., & Yero, I. G. (2020). Edge metric dimension of some graph operations. Bulletin of the Malaysian Mathematical Sciences Society, 43, 2465–2477.

[17] Prabhu, S., & Janany, T. J. (2026). Edge metric dimension of silicate networks. Communications in Combinatorics and Optimization, 11, 287–296.

[18] Pranata, P. Y., Kiftiah, M., & Fran, F. (2019). Jumlah bintang pada graf bunga matahari. Bimester Buletin Ilmiah Matematika Statistika dan Terapannya, 8, 447–455.

[19] Rahmadi, D., Robiultsani, M. F. N., & Abrori, M. (2025). On the edge metric dimension of helm and closed helm graph. Jurnal Matematika UNAND, 14, 232–239.

[20] Rajan, B., Rajasingh, I., & Venugopal, P. (2011). Metric dimension of uniform and quasi-uniform theta graphs. Journal of Computational Mathematical Sciences, 2(1), 37–46.

[21] Rinurwati, & Nabila, R. E. (2020). Metric and edge-metric dimensions of bobble-neighbourhood-corona graphs. Journal of Physics: Conference Series. https://iopscience.iop.org/article/10.1088/1742-6596/1836/1/012010

[22] Sharma, S. K., Raza, H., & Bhat, V. K. (2021). Computing edge metric dimension of one-pentagonal carbon nanocone. Frontiers in Physics, 9. https://doi.org/10.3389/fphy.2021.749166

[23] Singh, P., Sharma, S., Sharma, S. K., & Bhat, V. K. (2021). Metric dimension and edge metric dimension of windmill graphs. AIMS Mathematics, 6(9), 9138–9153.

[24] Wei, M., Fan, B., Lu, C., Yue, J., & Liu, J. (2026). The edge metric dimensions of convex polytopes. Discrete Applied Mathematics, 378, 294–306.

[25] Welyyanti, D., Arsyad, A., & Yulianti, L. (2023). Dimensi metrik amalgamasi graf theta. Journal of Mathematics and Its Applications. https://journal.its.ac.id/index.php/limits/article/view/5588