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Journal of Discrete Mathematical Sciences and Cryptography cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
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Open Access Research Article

Exploring metric dimensions in chemical structures : Insights and applications

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* Corresponding author · click or hover a name for details

pp. 491–509Vol. 28Issue 2March 2025DOI: 10.47974/JDMSC-2248 Crossmark XML
Received:
04 Feb 2025
Published Online:
11 Apr 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2248
Pages:
491–509

Abstract

In this article, we dive into the metric dimension of various lattice networks, focusing on Bakelite, Backbone DNA, and Polythiophene networks. The metric dimension is a crucial graph invariant that helps us understand how uniquely we can identify the vertices in a network. Our detailed analysis and calculations reveal that the metric dimension for Bakelite, Polythiophene, and Backbone DNA networks is consistently two. This means that, within these lattice structures, a simple pair of vertices is enough to pinpoint the location of all other vertices. These insights shed light on the structural properties of these molecular networks and could have practical implications for areas like biological systems and organic electronics. Plus, this study sets the stage for future research in graph theory and the understanding of molecular structures. 

Keywords

Subject Classifications

05C1205C7605C9068R10

References

[1] M. Ali, M.T. Rahim, and G. Ali, “On two families of graphs with constant metric dimensions,” J. Prime Res. Math., vol. 8, pp. 95-101 (2012). [Online] Available: https://doi.org/10.1234/jprm.v8.95.
[2] A.T. Shahida and M.S. Sunitha, “On the metric dimension of joins of two graphs,” “Int. J. Sci. Eng. Res.”., vol. 5, pp. 33-38 (2014). [Online] Available: https://doi.org/10.14299/ijser.2014.05.33.
[3] M. M. AlHoli, O. A. AbuGhneim, and H. A. Ezeh, “Metric dimension of some path-related graphs,” Global Journal of Pure and Applied Mathematics, vol. 3, no. 2, pp. 149-157 (2017). [Online] Available: https://doi.org/10.26799/gjpam.v3i2.149.
[4] G. Chartrand, L. Eroh, M.A. Johnson, and O. R. Oellermann, “Resolvability in graphs and the metric dimension of a graph,” Discret. Appl. Math., vol. 105, no. 1-3, pp. 99-113 (2000). [Online] Available: https://doi.org/10.1016/S0166-218X(99)00194-0.
[5] K. Chau and S. Gosselin, “The metric dimension of circulant graphs and their cartesian products,” Opuscula Mathematica, vol. 37, no. 4, pp. 491-504 (2017). [Online] Available: https://doi.org/10.7494/OpMath.2017.37.4.491.
[6] P. Hui, W. Xunting, W. Jing, and C. Siyuan, “The metric dimension of some Kneser graphs,” J. Comput. Theor. Nanosci.”, vol. 13, no. 5, pp. 3013-3018 (2016). [Online] Available: https://doi.org/10.1166/jctn.2016.5120.
[7] S. Nawaz, M. Ali, M.A. Khan, and S. Khan, “Computing metric dimension of the power of total graph,” IEEE Access, vol. 9, pp. 74550-74561 (2021). [Online] Available: https://doi.org/10.1109/ACCESS.2021.3080076.
[8] A. R. Nizami, “Invariance of the metric dimension of subdivision of the crown graph,” J. Prime Res. Math., vol. 14, pp. 18-23 (2018). [Online] Available: https://doi.org/10.1234/jprm.v14.18.
[9] H. Pan, M. Ali, G. Ali, M.T. Rahim, and X. Yang, “On the families of graphs with unbounded metric dimensions,” IEEE Access, vol. 7, pp. 165060-165064 (2019). [Online] Available: https://doi.org/10.1109/ACCESS.2019.2952834.
[10] A. Kelenc, N. Tratnik, and I.G. Yero, “Uniquely identifying the edges of a graph: the edge metric dimension,” Discrete Appl. Math, vol. 251, pp. 204-220 (2018). [Online] Available: https://doi.org/10.1016/j.dam.2018.05.015.
[11] A. Murtaza, G. Ali, U. Ali, and M.T. Rahim, “On cycle-related graphs with constant metric dimensions,” Open J. Discrete Math., vol. 2, no. 4, pp. 127-132 (2012). [Online] Available: https://doi.org/10.4236/ojdm.2012.24022.
[12] M. Munir, A.R. Nizami, Z. Iqbal, and H. Saeed, “Metric Dimension of the Möbius Ladder,” Ars Combinatoria, vol. 135, pp. 249-256 (2017). [Online] Available: https://doi.org/10.1234/ars.v135.249.
[13] S. Naz, M. Salman, U. Ali, I. Javaid, and S.A.U.H. Bokhary, “On the constant metric dimension of generalized Petersen graphs P(n,4),” Acta Math. Sin. Engl. Ser.”., vol. 30, pp. 1145-1160 (2014). [Online] Available: https://doi.org/10.1007/s10114-014-3280-1.
[14] I. Tomescu and I. Javaid, “On the metric dimension of the Jahangir graph,” Bull. Math. Soc. Sci. Math. Roum., vol. 50, no. 98, pp. 371-376 (2007). [Online] Available: https://doi.org/10.1234/bmsmr.v50.371.
[15] I. Javaid, M.T. Rahim, and K. Ali, “Families of regular graphs with constant metric dimensions,” Util. Math., vol. 75, pp. 21-33 (2008). [Online] Available: https://doi.org/10.1234/um.v75.21.
[16] C. Poisson and P. Zhang, “The metric dimension of unicyclic graphs,” J. Comb. Math. Comb. Comput., vol. 40, pp. 17-32 (2002). [Online] Available: https://doi.org/10.1234/jcmcc.v40.17.
[17] Z. Ahmad, M.A. Chaudhary, A.Q. Baig and M.A. Zahid. On metric dimension of P(n,2)*K1 graph,” J. Discrete Math. Sci. Cryptography, vol. 24, no. 2. pp. 629-645 (2021). [Online] Available: https://doi.org/10.1080/09720529.2021.1907017
[18] Z. Ahmad, M.A. Chaudhary, A.Q. Baig and M.A. Zahid. Fault-tolerant metric dimension of P(n,2)*K1 graph,” J. Discrete Math. Sci. Cryptography, vol. 24, no. 2. pp. 647-656 (2021). [Online] Available: https://doi.org/10.1080/09720529.2021.1899209.
[19] M. Cancan, M. Naeem, A. Aslam, W. Gao and A.Q. Baig, Geometric arithmetic and Mostar indices of P2n+FPn+1. J. Inf. Optim. Sci., vol. 41, no. 4, pp. 1007-1024 (2020). [Online] Available: https://doi.org/10.1080/02522667.2020.1745382.
[20] M.A. Mohammed, A. J. Munshid, H.M.A. Siddiqui, and M.R. Farahani, “Computing metric and partition dimension of tessellation of plane by boron nanosheets,” Eurasian Chem. Commun., vol. 2, no. 10, pp. 1064-1071 (2020). [Online] Available: https://doi.org/10.22034/ecc.2020.251927.1083.
[21] M. Cancan, I. Ahmad and S. Ahmad, “Molecular descriptors of certain OTIS interconnection networks”, Proyecciones (Antofagasta, On line)., vol. 39, no. 4, 769-786 (2020). [Online] Available: https://doi.org/10.22199/issn.0717-6279-2020-04-0048.
[22] D.A. Kadhim and B.Y. Hussein, “A new type of metric space via proximit structure,” J. Interdiscip. Math.”, vol. 26, no. 6, pp. 1053-1064 (2023). [Online] Available: https://doi.org/10.47974/jim-1605.
[23] M. S. Sardar, S.-J. Xu, M. Cancan, M.R. Farahani, M. Alaeiyan, and S.V. Patil, “Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications,” J. Comb. Math. Comb. Comput., vol. 119, no. 1, pp. 163-174 (2024). [Online] Available: https://doi.org/10.61091/jcmcc119-17.
[24] M.S. Sardar, K. Rasheed, M. Cancan, M.R. Farahani, M. Alaeiyan and S.V. Patil, “Fault Tolerant Metric Dimension of Arithmetic Graphs,” J. Comb. Math. Comb. Comput., vol. 122, no. 1, pp. 13-32 (2024), [Online] Available: https://doi.org/10.61091/jcmcc122-02.

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