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Topological indices of the subdivision of a family of partial cubes and computation of \(\mathrm{SiO}_2\) related structures. (English) Zbl 1421.92039

Summary: The aim of this paper is to apply the Djoković-Winkler relation to subdivisions of partial cubes and then to derive closed formulae for computing the topological indices of the subdivision graphs, provided the indices of its associated partial cubes are known. We have applied the obtained formulae to the subdivisions of circumcoronenes to compute the exact analytical expressions of its distance and degree-distance based indices. We have also obtained distance-based and degree-distance based indices of silicate graphs such as pruned quartz. Such silicate molecular structures have potential applications in nanomedicine for drug delivery systems, as these materials could serve as molecular belts for efficient drug delivery.

MSC:

92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
05C90 Applications of graph theory
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[1] M. Arockiaraj, J. Clement, K. Balasubramanian, Analytical expressions for topological properties of polycyclic benzenoid networks. J. Chemom. 30(11), 682-697 (2016) · doi:10.1002/cem.2851
[2] M. Arockiaraj, J. Clement, K. Balasubramanian, Topological indices and their applications to circumcised donut benzenoid systems, kekulenes and drugs. Polycycl. Aromat. Compd. (2017). https://doi.org/10.1080/10406638.2017.1411958 · doi:10.1080/10406638.2017.1411958
[3] M. Arockiaraj, S. Klavžar, S. Mushtaq, K. Balasubramanian, Distance-based topological indices of \[\text{ SiO }_2\] SiO2 nanosheets, nanotubes and nanotori. J. Math. Chem. 57(1), 343-369 (2019) · Zbl 1406.92735 · doi:10.1007/s10910-018-0956-8
[4] K. Balasubramanian, Applications of combinatorics and graph theory to spectrosocpy and quantum chemistry. Chem. Rev. 85(6), 599-618 (1985) · doi:10.1021/cr00070a005
[5] K. Balasubramanian, Cas scf/ci calculations on \[\text{ Si }_4\] Si4 and \[\text{ Si }_4^+\] Si4+. Chem. Phys. lett. 135(3), 283-287 (1987) · doi:10.1016/0009-2614(87)85157-6
[6] K. Balasubramanian, M. Randić, The characteristic polynomials of structures with pending bonds. Theor. Chim. Acta 61(4), 307-323 (1982) · doi:10.1007/BF00550410
[7] S.C. Basak, D. Mills, M.M. Mumtaz, K. Balasubramanian, Use of topological indices in predicting aryl hydrocarbon receptor binding potency of dibenzofurans: a hierarchical QSAR approach. Indian J. Chem. 42A(6), 1385-1391 (2003)
[8] D. Djoković, Distance preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14(3), 263-267 (1973) · Zbl 0245.05113 · doi:10.1016/0095-8956(73)90010-5
[9] I. Gutman, A.R. Ashrafi, The edge version of the Szeged index. Croat. Chem. Acta 81(2), 263-266 (2008)
[10] I. Gutman, S. Klavžar, An algorithm for the calculation of Szeged index of benzenoid hydrocarbons. J. Chem. Inf. Comput. Sci. 35(6), 1011-1014 (1995) · doi:10.1021/ci00028a008
[11] S. Hayat, M. Imran, Computation of topological indices of certain networks. Appl. Math. Comput. 240, 213-228 (2014) · Zbl 1334.05160
[12] A. Ilić, S. Klavžar, D. Stevanović, Calculating the degree distance of partial hamming graphs. MATCH Commun. Math. Comput. Chem. 63(2), 411-424 (2010) · Zbl 1265.05186
[13] A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani, The edge versions of the Wiener index. MATCH Commun. Math. Comput. Chem. 61(3), 663-672 (2009) · Zbl 1224.05144
[14] M. Javaid, M.U. Rehman, J. Cao, Topological indices of rhombus type silicate and oxide networks. Can. J. Chem. 95(2), 134-143 (2016) · doi:10.1139/cjc-2016-0486
[15] P.E. John, P.V. Khadikar, J. Singh, A method of computing the PI index of benzenoid hydrocarbons using orthogonal cuts. J. Math. Chem. 42(1), 37-45 (2007) · Zbl 1119.92073 · doi:10.1007/s10910-006-9100-2
[16] S.R.J. Kavitha, Topological Characterization of Certain Chemical Graphs, Ph.D. dissertation, University of Madras, India (2018)
[17] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, Another aspect of graph invariants depending on the path metric and an application in nanoscience. Comput. Math. Appl. 60(8), 2460-2468 (2010) · Zbl 1205.05232 · doi:10.1016/j.camwa.2010.08.042
[18] S. Klavžar, On the canonical metric representation, average distance, and partial Hamming graphs. Eur. J. Comb. 27(1), 68-73 (2006) · Zbl 1078.05028 · doi:10.1016/j.ejc.2004.07.008
[19] S. Klavžar, A bird’s eye view of the cut method and a survey of its recent applications in chemical graph theory. MATCH Commun. Math. Comput. Chem. 60(2), 255-274 (2008) · Zbl 1199.05316
[20] S. Klavžar, I. Gutman, B. Mohar, Labeling of benzenoid systems which reflects the vertex-distance relation. J. Chem. Inf. Comput. Sci. 35(3), 590-593 (1995) · doi:10.1021/ci00025a030
[21] S. Klavžar, I. Gutman, A. Rajapakse, Wiener numbers of pericondensed benzenoid hydrocarbons. Croat. Chem. Acta 70(4), 979-999 (1997)
[22] S. Klavžar, M.J. Nadjafi-Arani, Cut method: update on recent developments and equivalence of independent approaches. Curr. Org. Chem. 19(4), 348-358 (2015) · doi:10.2174/1385272819666141216232659
[23] J. Li, Y. Xiong, Z. Xie, X. Gao, J. Zhou, C. Yin, L. Tong, C. Chen, Z. Liu, J. Zhang, Template synthesis of an ultrathin \[\beta\] β-graphdiyne-like film using the Eglinton coupling reaction. ACS Appl. Mater. Interfaces 11(3), 2734-2739 (2018) · doi:10.1021/acsami.8b03028
[24] P. Liu, W. Long, Current mathematical methods used in QSAR/QSPR studies. Int. J. Mol. Sci. 10(5), 1978-1998 (2009) · doi:10.3390/ijms10051978
[25] J.B. Liu, S. Wang, C. Wang, S. Hayat, Further results on computation of topological indices of certain networks. IET Control Theory A 11(13), 2065-2071 (2017) · doi:10.1049/iet-cta.2016.1237
[26] P. Manuel, I. Rajasingh, M. Arockiaraj, Total-Szeged index of \[C_4\] C4-nanotubes, \[C_4\] C4-nanotori and denrimer nanostars. J. Comput. Theor. Nanosci. 10(2), 405-411 (2013) · doi:10.1166/jctn.2013.2712
[27] X. Niu, X. Mao, D. Yang, Z. Zhang, M. Si, D. Xue, Dirac cone in \[\alpha\] α-graphdiyne: a first-principles study. Nanoscale Res. Lett. 8(1), 469 (2013) · doi:10.1186/1556-276X-8-469
[28] D. Sundholm, L.N. Wirz, P. Schwerdtfeger, Novel hollow all-carbon structures. Nanoscale 7(38), 15886-15894 (2015) · doi:10.1039/C5NR04370K
[29] P. Winkler, Isometric embeddings in products of complete graphs. Discrete Appl. Math. 7(2), 221-225 (1984) · Zbl 0529.05055 · doi:10.1016/0166-218X(84)90069-6
[30] H. Yousefi-Azari, M.H. Khalifeh, A.R. Ashrafi, Calculating the edge-Wiener and Szeged indices of graphs. J. Comput. Appl. Math. 235(16), 4866-4870 (2011) · Zbl 1222.05037 · doi:10.1016/j.cam.2011.02.019
[31] C. Zhao, K. Balasubramanian, Geometries and spectroscopic properties of silicon clusters \[(\text{ Si }_5Si5, \text{ Si }_5^+\] Si5+, \[ \text{ Si }_5^-\] Si5-, \[ \text{ Si }_6 Si6, \text{ Si }_6^+\] Si6+, and \[\text{ Si }_6^-\] Si6-). J. Chem. Phys. 116(9), 3690-3699 (2002) · doi:10.1063/1.1446027
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