Computation of degree based topological aspects of 2D Kagome lattice M3C12X12
Mohammad Reza Farahanimohammad_farahani@mathdep.iust.ac.irSchool of Mathematics and Computer ScienceIran University of Science and Technology (IUST)Narmak, Tehran, 16844, Iran0000-0003-2969-4280View full profile → , Ahsan Aliahsan.ali@skt.umt.edu.pkDepartment of MathematicsUniversity of Management and TechnologySialkot Campus, Lahore, 51310, Pakistan0009-0000-3742-6230View full profile → , Anam Fiazanam.fiaz99@gmail.comDepartment of MathematicsUniversity of Management and TechnologySialkot Campus, Lahore, 51310, Pakistan0000-0002-2081-5595View full profile → , Mehran Azeemmalikmehran266@gmail.comDepartment of MathematicsUniversity of Management and TechnologySialkot Campus, Lahore, 51310, Pakistan0000-0002-0541-1277View full profile → , Shobha Veerangouda Patilshobhap49@gmail.comDepartment of MathematicsDr. M. S. Sheshgiri College of Engineering and TechnologyKLE Technological UniversityBelagavi, Karnataka, 590001, India0000-0002-6210-832XView full profile → , *Mehdi AlaeiyanCorresponding authoralaeiyan@iust.ac.irSchool of Mathematics and Computer ScienceIran University of Science and Technology (IUST)Narmak, Tehran, 16844, Iran0000-0003-2185-5967View full profile → , Murat Cancanmcancan@yyu.edu.trDepartment of MathematicsFaculty of EducationVan Yüzüncü Yıl UniversityVan, 65090, Turkey0000-0002-8606-2274View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Sep 2024
- Published Online:
- 31 Jul 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2452
- Pages:
- 2787–2805
Abstract
To connect the biological activity of quantitative structure - activity (QSAR) and quantitative structure - property (QSPR) interactions, topological indices are employed in structural underpinnings with their physical attributes, such as strain energy, chromatic aberration, boiling point, melting point, stability, etc. Such projections can be completed quickly and easily, degree - based topological indices are in the middle of the many types of descriptors in this research. In this study, we calculate the 2D structures of M3C12X12 lattice. Additionally, we compute the general Randic index, first and second Zagreb indices, atom - bond connectivity, geometric arithmetic index multiple geometric - arithmetic index and multiple atom - bond connectivity. We also give the exact formulas for the 5th iteration of the GA index and 4th iteration of the ABC index for these chemical graphs.
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References
[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science, vol. 306, no. 5696, pp. 666–669 (2004), doi: 10.1126/science.1102896.
[2] M. R. Lukatskaya, O. Mashtalir, C. E. Ren, Y. Dall’Agnese, P. Rozier, P. L. Taberna, M. Naguib, P. Simon, M. W. Barsoum, and Y. Gogotsi, “Cation intercalation and high volumetric capacitance of two-dimensional titanium carbide,” Science, vol. 341, no. 6153, pp. 1502–1505 (2013), doi: 10.1126/science.1241488.
[3] M. Corso, W. Auwärter, M. Muntwiler, A. Tamai, T. Greber, and J. Osterwalder, “Boron nitride nanomesh,” Science, vol. 303, no. 5655, pp. 217–220 (2004), doi: 10.1126/science.1091979.
[4] P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, and B. Ealet, “Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon,” Phys. Rev. Lett., vol. 108, no. 15, p. 155501 (2012), doi: 10.1103/PhysRevLett.108.155501.
[5] S. Y. Xie, X. B. Li, Y. Y. Sun, Y. L. Zhang, D. Han, W. Q. Tian, W. Q. Wang, Y. S. Zheng, S. B. Zhang, and H. B. Sun, “Theoretical characterization of reduction dynamics for graphene oxide by alkaline-earth metals,” Carbon, vol. 52, pp. 122–127 (2013), doi: 10.1016/j.carbon.2012.09.012.
[6] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, “Atomically thin MoS2: A new direct-gap semiconductor,” Phys. Rev. Lett., vol. 105, no. 13, p. 136805 (2010), doi: 10.1103/PhysRevLett.105.136805.
[7] S. Jungthawan, P. Reunchan, and S. Limpijumnong, “Theoretical study of strained porous graphene structures and their gas separation properties,” Carbon, vol. 54, pp. 359–364 (2013), doi: 10.1016/j.carbon.2012.11.048.
[8] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, “Single-layer MoS2 transistors,” Nat. Nanotechnol., vol. 6, no. 3, pp. 147–150 (2011), doi: 10.1038/nnano.2010.279.
[9] N. Papasimakis, S. Thongrattanasiri, N. I. Zheludev, and F. J. García de Abajo, “The magnetic response of graphene split-ring metamaterials,” Light Sci. Appl., vol. 2, no. 7, p. e78 (2013), doi: 10.1038/lsa.2013.34.
[10] T. Kambe, R. Sakamoto, K. Hoshiko, K. Takada, M. Miyachi, J. H. Ryu, S. Sasaki, J. Kim, K. Nakazato, and M. Takata, “π-Conjugated nickel bis(dithiolene) complex nanosheet,” J. Am. Chem. Soc., vol. 135, no. 7, pp. 2462–2465 (2013), doi: 10.1021/ja312380b.
[11] A. Aslam, M. K. Jamil, W. Gao, and W. Nazeer, “Topological aspects of some dendrimer structures,” Nanotechnol. Rev., vol. 7, no. 2, pp. 123–129 (2018), doi: 10.1515/ntrev-2017-0184.
[12] W. Gao, M. Aamir, Z. Iqbal, M. Ishaq, and A. Aslam, “On irregularity measures of some dendrimer structures,” Mathematics, vol. 7, no. 3, p. 271 (2019), doi: 10.3390/math7030271.
[13] A. J. Clough, J. W. Yoo, M. H. Mecklenburg, and S. C. Marinescu, “Two-dimensional metal-organic surfaces for efficient hydrogen evolution from water,” J. Am. Chem. Soc., vol. 137, no. 1, pp. 118–121 (2015), doi: 10.1021/ja5116937.
[14] X. Li and I. Gutman, Mathematical Aspects of Randic-Type Molecular Structure Descriptors. Kragujevac, Serbia: Univ. Kragujevac (2006).
[15] A. Aslam, Y. Bashir, S. Ahmad, and W. Gao, “On topological indices of certain dendrimer structures,” Z. Naturforsch. A, vol. 72, no. 6, pp. 559–566 (2017), doi: 10.1515/zna-2017-0081.
[16] Y. Bashir, A. Aslam, M. Kamran, I. Qureshi, A. Jahangir, M. Rafiq, N. Bibi, and N. Muhammad, “On forgotten topological indices of some dendrimer structures,” Molecules, vol. 22, no. 6, p. 867 (2017), doi: 10.3390/molecules22060867.
[17] W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, M. Aamir, and M. A. Binyamin, “Bounds on topological descriptors of the corona product of F-sum of connected graphs,” IEEE Access, vol. 7, pp. 26788–26796 (2019), doi: 10.1109/ACCESS.2019.2900061.
[18] W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, and R. Sarfraz, “Topological aspects of dendrimers via distance based descriptors,” IEEE Access, vol. 7, pp. 35619–35630 (2019), doi: 10.1109/ACCESS.2019.2904736.
[19] W. Gao, M. K. Siddiqui, M. Naeem, and M. Imran, “Computing multiple ABC index and multiple GA index of some grid graphs,” Open Phys., vol. 16, no. 1, pp. 588–598 (2018), doi: 10.1515/phys-2018-0077.
[20] Z. Iqbal, M. Ishaq, and M. Aamir, “On eccentricity-based topological descriptors of dendrimers,” Iran. J. Sci. Technol. Trans. A Sci., vol. 43, no. 4, pp. 1523–1533 (2019), doi: 10.1007/s40995-018-0621-x.
[21] Z. Iqbal, A. Aslam, M. Ishaq, and M. Aamir, “Characteristic study of irregularity measures of some nanotubes,” Can. J. Phys., vol. 97, no. 10, pp. 1125–1132 (2019), doi: 10.1139/cjp-2018-0619.
[22] Z. Iqbal, A. Aslam, M. Ishaq, and W. Gao, “The edge versions of degree-based topological descriptors of dendrimers,” J. Clust. Sci., vol. 31, no. 2, pp. 445–452 (2020), doi: 10.1007/s10876-019-01658-w.
[23] H. Yang, M. Imran, S. Akhter, Z. Iqbal, and M. K. Siddiqui, “On distance-based topological descriptors of subdivision vertex-edge join of three graphs,” IEEE Access, vol. 7, pp. 143381–143391 (2019), doi: 10.1109/ACCESS.2019.2944860.
[24] D. Zhao, Z. Iqbal, R. Irfan, M. A. Chaudhry, M. Ishaq, M. K. Jamil, and A. Fahad, “Comparison of irregularity indices of several dendrimer structures,” Processes, vol. 7, no. 10, p. 662 (2019), doi: 10.3390/pr7100662.
[25] J. Zheng, Z. Iqbal, A. Fahad, A. Zafar, A. Aslam, M. I. Qureshi, and R. Irfan, “Some eccentricity-based topological indices and polynomials of poly(ethyleneamidoamine) (PETAA) dendrimers,” Processes, vol. 7, no. 7, p. 433 (2019), doi: 10.3390/pr7070433.
[26] Q. Zhou, J. Wang, T. S. Chwee, G. Wu, X. Wang, Q. Ye, J. Xu, and S. W. Yang, “Topological insulators based on 2D shape-persistent organic ligand complexes,” Nanoscale, vol. 7, no. 2, pp. 727–735 (2015), doi: 10.1039/C4NR05247A.
[27] X. Zhang, H. G. G. Reddy, A. Usha, M. C. Shanmukha, M. R. Farahani, and M. Alaeiyan, “A study on anti-malaria drugs using degree-based topological indices through QSPR analysis,” Math. Biosci. Eng., vol. 20, no. 2, pp. 3594–3609 (2023), doi: 10.3934/mbe.2023167.
[28] D. Afzal, F. Afzal, M. R. Farahani, and S. Ali, “On computation of recently defined degree-based topological indices of some families of convex polytopes via M-polynomial,” Complexity, vol. 2021, pp. 1–11 (2021), doi: 10.1155/2021/5881476.
[29] F. Chaudhry, I. Shoukat, D. Afzal, C. Park, M. Cancan, and M. R. Farahani, “M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide,” J. Chem., vol. 2021, Art. no. 6679819, pp. 1–12 (2021), doi: 10.1155/2021/6679819.
[30] M. Alaeiyan, “Characteristics and eigenvalues of the newly defined Ala graph,” Phys. Scr., vol. 100, no. 5, Art. no. 055201 (2025), doi: 10.1088/1402-4896/adc3d2.
[31] M. Imran, M. R. Farahani, M. Cancan, M. Alaeiyan, and A. Akgül, “On topological indices of certain families of graphs,” Phys. Scr., vol. 100, no. 1 (2024), doi: 10.1088/1402-4896/ad9065.
[32] D. Sheberla, L. Sun, M. A. Blood-Forsythe, E. Er, C. R. Wade, C. K. Brozek, A. Aspuru-Guzik, and M. Dincă, “High electrical conductivity in Ni3(2,3,6,7,10,11-hexaiminotriphenylene)2, a semiconducting metal–organic graphene analogue,” J. Am. Chem. Soc., vol. 136, no. 25, pp. 8859–8862 (2014), doi: 10.1021/ja502765n.
[33] C. Chakravarty, B. Mandal, and P. Sarkar, “Coronene-based metal-organic framework: A theoretical exploration,” Phys. Chem. Chem. Phys., vol. 18, no. 36, pp. 25277–25283 (2016), doi: 10.1039/C6CP05495A.
[34] P. Wang, X. Jiang, J. Hu, B. Wang, T. Zhou, H. Yuan, and J. Zhao, “Robust spin manipulation in 2D organometallic Kagome lattices: A first-principles study,” Phys. Chem. Chem. Phys., vol. 22, no. 19, pp. 11045–11052 (2020), doi: 10.1039/D0CP00742K.
