[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Edge resolvability of generalized honeycomb rhombic torus

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Minimum resolving sets (edge or vertex) have become integral to computer science, molecular topology, and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for uniquely identifying each item in the network. The metric(respectively edge metric) dimension of a graph is the smallest number of the nodes needed to determine all other nodes (resp. edges) based on shortest path distances uniquely. Metric and edge metric dimensions as graph invariants have numerous applications, including robot navigation, pharmaceutical chemistry, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. A honeycomb torus network can be obtained by joining pairs of nodes of degree two of the honeycomb mesh. Honeycomb torus has recently gained recognition as an attractive alternative to existing torus interconnection networks in parallel and distributed applications. In this article, we will discuss the Honeycomb Rhombic torus graph on the basis of edge metric dimension.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Abbas, S., Raza, Z., Siddiqui, N., Khan, F., Whangbo, T.: Edge metric dimension of honeycomb and hexagonal networks for IoT. Comput., Mater. Continua 71(2), 2683–2695 (2022)

    Article  MATH  Google Scholar 

  2. Ahsan, M., Zahid, Z., Zafar, S., Rafiq, A., Sindhu, M.S., Umar, M.: Computing the edge metric dimension of convex polytopes related graphs. J. Math. Comput. Sci 22(2), 174–188 (2021)

    Article  MATH  Google Scholar 

  3. Bukhari, S., Jamil, M.K., Azeem, M., Swaray, S.: Patched network and its vertex-edge metric-based dimension. IEEE Access 11, 4478–4485 (2023)

    Article  MATH  Google Scholar 

  4. Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discret. Appl. Math. 105(1–3), 99–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cho, H.J., Hsu, L.Y.: Generalized honeycomb torus. Inf. Process. Lett. 86(4), 185–190 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Garey, M.R., Johnson, D.S.: A Guide to the Theory of NP-Completeness. Computers and intractability, pp. 37-79 (1990)

  7. Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2(1), 191–195 (1976)

    MathSciNet  MATH  Google Scholar 

  8. Kelenc, A., Tratnik, N., Yero, I.G.: Uniquely identifying the edges of a graph: the edge metric dimension. Discret. Appl. Math. 251, 204–220 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Knor, M., Majstorović, S., Toshi, A.T.M., Škrekovski, R., Yero, I.G.: Graphs with the edge metric dimension smaller than the metric dimension. Appl. Math. Comput. 401, 126076 (2021). https://doi.org/10.1016/j.amc.2021.126076

    Article  MathSciNet  MATH  Google Scholar 

  10. Koam, A.N., Ahmad, A.: Barycentric subdivision of Cayley graphs with constant edge metric dimension. IEEE Access 8, 80624–80628 (2020)

    Article  MATH  Google Scholar 

  11. Manuel, P., Bharati, R., Rajasingh, I.: On minimum metric dimension of honeycomb networks. J. Discrete Algorithms 6(1), 20–27 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Peterin, I., Yero, I.G.: Edge metric dimension of some graph operations. Bull. Malays. Math. Sci. Soc. 43(3), 2465–2477 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Raza, Z., Bataineh, M.S.: The comparative analysis of metric and edge metric dimension of some subdivisions of the wheel graph. Asian-European J. Math. 14(04), 1–14 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rosenfeld, A., Raghavachari, B., Khuller, S.: Landmarks in graphs. Discr. Appl. Math. 70, 217–229 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sebő, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sharma, K., Bhat, V.K., Sharma, S.K.: Edge metric dimension and edge basis of one-heptagonal carbon nanocone networks. IEEE Access 10, 29558–29566 (2022)

    Article  MATH  Google Scholar 

  17. Sharma, S.K., Bhat, V.K.: Metric dimension of heptagonal circular ladder. Discrete Math., Algorith. Appl. 13(01), 2050095 (2021). https://doi.org/10.1142/S1793830920500950

    Article  MathSciNet  MATH  Google Scholar 

  18. Sharma, S.K., Bhat, V.K., Raza, H., Sharma, K.: Metric and edge metric dimension of zigzag edge coronoid fused with starphene. arXiv preprint arXiv:2107.14484 (2021)

  19. Siddiqui, H.M.A., Imran, M.: Computing the metric dimension of wheel related graphs. Appl. Math. Comput. 242, 624–632 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Singh, M., Bhat, V.K.: On metric dimension of hendecagonal circular ladder \( H_ n \). Ann. Univ. Craiova-Math. Comput. Sci. Ser. 50(2), 394–403 (2023)

    MATH  Google Scholar 

  21. Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)

    MathSciNet  MATH  Google Scholar 

  22. Stojmenovic, I.: Honeycomb networks: topological properties and communication algorithms. IEEE Trans. Parallel Distrib. Syst. 8(10), 1036–1042 (1997)

    Article  MATH  Google Scholar 

  23. Yang, B., Rafiullah, M., Siddiqui, H.M.A., Ahmad, S.: On resolvability parameters of some wheel-related graphs. J. Chem. 2019, 1–9 (2019)

    MATH  Google Scholar 

  24. Zhang, Y., Gao, S.: On the edge metric dimension of convex polytopes and its related graphs. J. Comb. Optim. 39(2), 334–350 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zubrilina, N.: On the edge dimension of a graph. Discret. Math. 341(7), 2083–2088 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

No funding available to conduct this. The authors declare no conflict of interest.

Author information

Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan

    Ayesha Andalib Kiran & Hani Shaker

  2. Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia

    Suhadi Wido Saputro

Authors
  1. Ayesha Andalib Kiran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hani Shaker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Suhadi Wido Saputro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hani Shaker.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiran, A.A., Shaker, H. & Saputro, S.W. Edge resolvability of generalized honeycomb rhombic torus. J. Appl. Math. Comput. 71, 303–322 (2025). https://doi.org/10.1007/s12190-024-02231-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-024-02231-z

Keywords

Mathematics Subject Classification