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doi: 10.3934/dcdss.2020418

Topological indices of discrete molecular structure

1. 

Department of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, Pakistan

2. 

Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, 30203-Cartagena, Spain

3. 

Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, 57000, Pakistan

* Corresponding author: Muhammad Aamer Rashid

Received  April 2019 Revised  May 2020 Published  August 2020

Topological indices defined on molecular structures can help researchers better understand the physical features, chemical reactivity, and biological activity. Thus, the study of the topological indices on chemical structure of chemical materials and drugs can make up for lack of chemical experiments and can provide a theoretical basis for the manufacturing of drugs and chemical materials. In this paper, we focus on the family of smart polymer which is widely used in anticancer drugs manufacturing. In chemical graph theory, a topological index is a numerical representation of a chemical structure which correlates certain physico-chemical characteristics of underlying chemical compounds e.g., boiling point and melting point. More preciously, we focus on the family of smart polymer which is widely used in anticancer drugs manufacturing, and computed exact results for degree based topological indices.

Citation: Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui, Juan L. G. Guirao, Najma Abdul Rehman. Topological indices of discrete molecular structure. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020418
References:
[1]

A. T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett., 89 (1982), 399-404.  doi: 10.1016/0009-2614(82)80009-2.  Google Scholar

[2]

A. T. Balaban and L. V. Quintas, The smallest graphs, trees, and $4$-trees with degenerate topological index, J. Math. Chem., 14 (1983), 213-233.   Google Scholar

[3]

A. M. ButtM. C. IqbalM. Amin and H. Katas, Synergistic effect of $pH-$responsive folate functionalized poloxamer $TPGS-$mixed micelles on targeted delivery of anticancer drugs, International Journal of Nanomedicine, 10 (2015), 1321-1334.   Google Scholar

[4]

A. ChonkarU. Nayak and N. Udupa, Smart polymers in nasal drug delivery, Indian Journal of Pharmaceutical Sciences, 77 (2015), 367-375.   Google Scholar

[5]

T. Doslic, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp., 1 (2008), 66-80.  doi: 10.26493/1855-3974.15.895.  Google Scholar

[6]

A. Duro-CastanoJ. Movellan and M. J. Vicent, Smart branched polymer drug conjugates as nano-sized drug delivery systems, Biomaterials Science, 3 (2015), 1321-1334.  doi: 10.1039/C5BM00166H.  Google Scholar

[7]

M. EliasiA. Iranmanesh and I. Gutman, Multiplicative version of first zagreb index, MATCH Commun. Math. Comput. Chem., 68 (2012), 217-230.   Google Scholar

[8]

B. FurtulaA. Graovac and D. Vukičević, Augmented zagreb index, J. Math. Chem., 48 (2010), 370-380.  doi: 10.1007/s10910-010-9677-3.  Google Scholar

[9]

B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem., 53 (2015), 1184-1190.  doi: 10.1007/s10910-015-0480-z.  Google Scholar

[10]

W. Gao, M. K. Siddiqui, M. Imran, M. K. Jamil and M. R. Farahani, Forgotten topological index of Chemical Structure in Drugs, Saudi Pharmaceutical Journal, 24 (2016), 258–267. doi: 10.1016/j.jsps.2016.04.012.  Google Scholar

[11]

W. Gao and M. K. Siddiqui, Molecular descriptors of nanotube, oxide, silicate, and triangulene networks, Journal of Chemistry, 2017 (2017), 1–10. doi: 10.1155/2017/6540754.  Google Scholar

[12]

W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22(9) (2017), 1496–1507. doi: 10.3390/molecules22091496.  Google Scholar

[13]

W. GaoM. K. SiddiquiM. Naeem and M. Imran, Computing multiple ABC index and multiple GA index of some grid graphs, Open. Phy., 16 (2018), 588-598.  doi: 10.1515/phys-2018-0077.  Google Scholar

[14]

M. Ghorbani an N. Azimi, Note on multiple Zagreb indices, Iran. J. Math. Chem., 3 (2012), 137-143.   Google Scholar

[15]

I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 50 (2004), 83-92.   Google Scholar

[16]

I. Gutman and N. Trinajst$\acute{c}$, Graph theory and molecular orbitals., Total $\pi$-electron energy of alternant hydrocarbons, Chemical Physics Letters, 17 (1972), 535-538.  doi: 10.1016/0009-2614(72)85099-1.  Google Scholar

[17]

I. GutmanB. FurtulaZ. K. Vukićević and G. Popivoda, On Zagreb Indices and Coindices, MATCH Commun. Math. Comput. Chem., 74 (2015), 5-16.   Google Scholar

[18]

N. T. M. Hai and P. Broekmann, Smart hybrid polymers for advanced damascene electroplating: combination of superfll and leveling properties, Chem Electro Chem, 2 (2015) 1096–1099. doi: 10.1002/celc.201500104.  Google Scholar

[19]

M. HrubýS. K. Filippov and P. Štěpánek, Smart polymers in drug delivery systems on crossroads: Which way deserves following, European Polymer Journal, 65 (2015), 82-97.  doi: 10.1016/j.eurpolymj.2015.01.016.  Google Scholar

[20]

N. Idrees, M. N. Naeem, F. Hussain, A. Sadiq and M. K. Siddiqui, Molecular Descriptors of Benzenoid System, Quimica Nova., 40 (2017), 143–145. doi: 10.21577/0100-4042.20160173.  Google Scholar

[21]

S. M. KangM. K. SiddiquiN. A. RehmanM. Naeem and M. H. Muhammad, Topological properties of 2-dimensional silicon-carbons, IEEE Access., 6 (2018), 59362-59373.  doi: 10.1109/ACCESS.2018.2874461.  Google Scholar

[22]

A. R. KatritzkyR. JainA. LomakaR. PetrukhinU. Maran and M. Karelson, Perspective on the relationship between melting points and chemical structure, Crystal Growth & Design, 1 (2001), 261-265.  doi: 10.1021/cg010009s.  Google Scholar

[23]

J. Khandare and M. Calderon, Dendritic polymers for smart drug delivery applications, Nanoscale, 7 (2015), 3806-3807.  doi: 10.1039/C5NR90030A.  Google Scholar

[24]

M. KnorR. škrekovski and A. Tepeh, Convexity result and trees with large Balaban index, Applied Mathematics and Nonlinear Sciences, 3 (2018), 433-445.  doi: 10.21042/AMNS.2018.2.00034.  Google Scholar

[25]

A. KroningA. Furchner and D. Aulich, In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment, ACS Applied Materials & Interfaces, 7 (2015), 12430-12439.  doi: 10.1021/am5075997.  Google Scholar

[26]

V. Lokeshav, T. Deepika, P. S. Ranjini and I. N. Cangul, Operations of Nano structures via SDD, $ABC_{4}$ and $GA_{5}$ indices, Applied Mathematics and Nonlinear Sciences, 2 (2017), 173-180. doi: 10.21042/AMNS.2017.1.00014.  Google Scholar

[27]

N. MarquesA. M. Maia and R. C. Balaban, Development of dual-sensitive smart polymers by grafing chitosan with poly (N-isopropylacrylamide): An overview, Polimeros, 25 (2015), 237-246.  doi: 10.1590/0104-1428.1744.  Google Scholar

[28]

N. Nishiyama and K. Kataoka, Polymeric micelle drug carrier systems: PEG-PAsp(Dox) and second generation of micellar drugs, in Polymer Drugs in the Clinical Stage, Advances in Experimental Medicine and Biology, 519 (2003), 155-177.   Google Scholar

[29]

K. OsadaR. J. Christie and K. Kataoka, Polymeric micelles from poly(ethylene glycol)-poly(amino acid) block copolymer for drug and gene delivery, Journal of the Royal Society Interface, 6 (2009), 325-339.  doi: 10.1098/rsif.2008.0547.focus.  Google Scholar

[30]

P. S. Ranjini, V. Lokesha and A. Usha, Relation between phenylene and hexagonal squeez using harmonic index, Int J Graph Theory, 1 (2013), 116–21. Google Scholar

[31]

K. ShanthiK. VimalaD. Gopi and S. Kannan, Fabrication of a pH responsive DOX conjugated PEGylated palladium nanoparticle mediated drug delivery system: an in vitro and in vivo evaluation, RSC Advances, 5 (2015), 44998-45014.  doi: 10.1039/C5RA05803A.  Google Scholar

[32]

M. K. Siddiqui, M. Imran and A. Ahmad, On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput., 280 (2016), 132–139. doi: 10.1016/j.amc.2016.01.041.  Google Scholar

[33]

M. K. Siddiqui, M. Naeem, N. A. Rahman and M. Imran, Computing topological indicesof certain networks, J. Optoelectron. Adv. Mater., 18 (2016), 884–892. Google Scholar

[34]

H. J. Wiener, Structural determination of parafn boiling points, Journal of the American Chemical Society, 69 (1947), 17-20.  doi: 10.1021/ja01193a005.  Google Scholar

show all references

References:
[1]

A. T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett., 89 (1982), 399-404.  doi: 10.1016/0009-2614(82)80009-2.  Google Scholar

[2]

A. T. Balaban and L. V. Quintas, The smallest graphs, trees, and $4$-trees with degenerate topological index, J. Math. Chem., 14 (1983), 213-233.   Google Scholar

[3]

A. M. ButtM. C. IqbalM. Amin and H. Katas, Synergistic effect of $pH-$responsive folate functionalized poloxamer $TPGS-$mixed micelles on targeted delivery of anticancer drugs, International Journal of Nanomedicine, 10 (2015), 1321-1334.   Google Scholar

[4]

A. ChonkarU. Nayak and N. Udupa, Smart polymers in nasal drug delivery, Indian Journal of Pharmaceutical Sciences, 77 (2015), 367-375.   Google Scholar

[5]

T. Doslic, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp., 1 (2008), 66-80.  doi: 10.26493/1855-3974.15.895.  Google Scholar

[6]

A. Duro-CastanoJ. Movellan and M. J. Vicent, Smart branched polymer drug conjugates as nano-sized drug delivery systems, Biomaterials Science, 3 (2015), 1321-1334.  doi: 10.1039/C5BM00166H.  Google Scholar

[7]

M. EliasiA. Iranmanesh and I. Gutman, Multiplicative version of first zagreb index, MATCH Commun. Math. Comput. Chem., 68 (2012), 217-230.   Google Scholar

[8]

B. FurtulaA. Graovac and D. Vukičević, Augmented zagreb index, J. Math. Chem., 48 (2010), 370-380.  doi: 10.1007/s10910-010-9677-3.  Google Scholar

[9]

B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem., 53 (2015), 1184-1190.  doi: 10.1007/s10910-015-0480-z.  Google Scholar

[10]

W. Gao, M. K. Siddiqui, M. Imran, M. K. Jamil and M. R. Farahani, Forgotten topological index of Chemical Structure in Drugs, Saudi Pharmaceutical Journal, 24 (2016), 258–267. doi: 10.1016/j.jsps.2016.04.012.  Google Scholar

[11]

W. Gao and M. K. Siddiqui, Molecular descriptors of nanotube, oxide, silicate, and triangulene networks, Journal of Chemistry, 2017 (2017), 1–10. doi: 10.1155/2017/6540754.  Google Scholar

[12]

W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22(9) (2017), 1496–1507. doi: 10.3390/molecules22091496.  Google Scholar

[13]

W. GaoM. K. SiddiquiM. Naeem and M. Imran, Computing multiple ABC index and multiple GA index of some grid graphs, Open. Phy., 16 (2018), 588-598.  doi: 10.1515/phys-2018-0077.  Google Scholar

[14]

M. Ghorbani an N. Azimi, Note on multiple Zagreb indices, Iran. J. Math. Chem., 3 (2012), 137-143.   Google Scholar

[15]

I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 50 (2004), 83-92.   Google Scholar

[16]

I. Gutman and N. Trinajst$\acute{c}$, Graph theory and molecular orbitals., Total $\pi$-electron energy of alternant hydrocarbons, Chemical Physics Letters, 17 (1972), 535-538.  doi: 10.1016/0009-2614(72)85099-1.  Google Scholar

[17]

I. GutmanB. FurtulaZ. K. Vukićević and G. Popivoda, On Zagreb Indices and Coindices, MATCH Commun. Math. Comput. Chem., 74 (2015), 5-16.   Google Scholar

[18]

N. T. M. Hai and P. Broekmann, Smart hybrid polymers for advanced damascene electroplating: combination of superfll and leveling properties, Chem Electro Chem, 2 (2015) 1096–1099. doi: 10.1002/celc.201500104.  Google Scholar

[19]

M. HrubýS. K. Filippov and P. Štěpánek, Smart polymers in drug delivery systems on crossroads: Which way deserves following, European Polymer Journal, 65 (2015), 82-97.  doi: 10.1016/j.eurpolymj.2015.01.016.  Google Scholar

[20]

N. Idrees, M. N. Naeem, F. Hussain, A. Sadiq and M. K. Siddiqui, Molecular Descriptors of Benzenoid System, Quimica Nova., 40 (2017), 143–145. doi: 10.21577/0100-4042.20160173.  Google Scholar

[21]

S. M. KangM. K. SiddiquiN. A. RehmanM. Naeem and M. H. Muhammad, Topological properties of 2-dimensional silicon-carbons, IEEE Access., 6 (2018), 59362-59373.  doi: 10.1109/ACCESS.2018.2874461.  Google Scholar

[22]

A. R. KatritzkyR. JainA. LomakaR. PetrukhinU. Maran and M. Karelson, Perspective on the relationship between melting points and chemical structure, Crystal Growth & Design, 1 (2001), 261-265.  doi: 10.1021/cg010009s.  Google Scholar

[23]

J. Khandare and M. Calderon, Dendritic polymers for smart drug delivery applications, Nanoscale, 7 (2015), 3806-3807.  doi: 10.1039/C5NR90030A.  Google Scholar

[24]

M. KnorR. škrekovski and A. Tepeh, Convexity result and trees with large Balaban index, Applied Mathematics and Nonlinear Sciences, 3 (2018), 433-445.  doi: 10.21042/AMNS.2018.2.00034.  Google Scholar

[25]

A. KroningA. Furchner and D. Aulich, In situ infrared ellipsometry for protein adsorption studies on ultrathin smart polymer brushes in aqueous environment, ACS Applied Materials & Interfaces, 7 (2015), 12430-12439.  doi: 10.1021/am5075997.  Google Scholar

[26]

V. Lokeshav, T. Deepika, P. S. Ranjini and I. N. Cangul, Operations of Nano structures via SDD, $ABC_{4}$ and $GA_{5}$ indices, Applied Mathematics and Nonlinear Sciences, 2 (2017), 173-180. doi: 10.21042/AMNS.2017.1.00014.  Google Scholar

[27]

N. MarquesA. M. Maia and R. C. Balaban, Development of dual-sensitive smart polymers by grafing chitosan with poly (N-isopropylacrylamide): An overview, Polimeros, 25 (2015), 237-246.  doi: 10.1590/0104-1428.1744.  Google Scholar

[28]

N. Nishiyama and K. Kataoka, Polymeric micelle drug carrier systems: PEG-PAsp(Dox) and second generation of micellar drugs, in Polymer Drugs in the Clinical Stage, Advances in Experimental Medicine and Biology, 519 (2003), 155-177.   Google Scholar

[29]

K. OsadaR. J. Christie and K. Kataoka, Polymeric micelles from poly(ethylene glycol)-poly(amino acid) block copolymer for drug and gene delivery, Journal of the Royal Society Interface, 6 (2009), 325-339.  doi: 10.1098/rsif.2008.0547.focus.  Google Scholar

[30]

P. S. Ranjini, V. Lokesha and A. Usha, Relation between phenylene and hexagonal squeez using harmonic index, Int J Graph Theory, 1 (2013), 116–21. Google Scholar

[31]

K. ShanthiK. VimalaD. Gopi and S. Kannan, Fabrication of a pH responsive DOX conjugated PEGylated palladium nanoparticle mediated drug delivery system: an in vitro and in vivo evaluation, RSC Advances, 5 (2015), 44998-45014.  doi: 10.1039/C5RA05803A.  Google Scholar

[32]

M. K. Siddiqui, M. Imran and A. Ahmad, On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput., 280 (2016), 132–139. doi: 10.1016/j.amc.2016.01.041.  Google Scholar

[33]

M. K. Siddiqui, M. Naeem, N. A. Rahman and M. Imran, Computing topological indicesof certain networks, J. Optoelectron. Adv. Mater., 18 (2016), 884–892. Google Scholar

[34]

H. J. Wiener, Structural determination of parafn boiling points, Journal of the American Chemical Society, 69 (1947), 17-20.  doi: 10.1021/ja01193a005.  Google Scholar

Figure 1.  (a) SP[n]    (b) SP[1]
Figure 2.  (a) SP[2]    (b) SP[3]
Figure 3.  (a) $ \overline{M_{1}(G)}(red) $, $ \overline{M_{2}(G)}(blue) $, (b) F(G)(red), AZI(G)(blue), J(G)(green), ABC(G)(brown)
Table 1.  Numerical computation of all indices for $ SP[n] $
$ n $ $ \overline{M_{1}(G)} $ $ \overline{M_{2}(G)} $ $ F(G) $ $ AZI(G) $ $ J(G) $, $ ABC(G) $
$ 1 $ $ 6082 $ $ 6467 $ $ 778 $ $ 468.4 $ $ 257.2 $ $ 42.27 $
$ 2 $ $ 22716 $ $ 24572 $ $ 1522 $ $ 902.5 $ $ 507.9 $ $ 80.87 $
$ 3 $ $ 49934 $ $ 54341 $ $ 2266 $ $ 1337 $ $ 759.1 $ $ 119.5 $
$ 4 $ $ 87736 $ $ 95776 $ $ 3010 $ $ 1771 $ $ 1010 $ $ 158.2 $
$ n $ $ \overline{M_{1}(G)} $ $ \overline{M_{2}(G)} $ $ F(G) $ $ AZI(G) $ $ J(G) $, $ ABC(G) $
$ 1 $ $ 6082 $ $ 6467 $ $ 778 $ $ 468.4 $ $ 257.2 $ $ 42.27 $
$ 2 $ $ 22716 $ $ 24572 $ $ 1522 $ $ 902.5 $ $ 507.9 $ $ 80.87 $
$ 3 $ $ 49934 $ $ 54341 $ $ 2266 $ $ 1337 $ $ 759.1 $ $ 119.5 $
$ 4 $ $ 87736 $ $ 95776 $ $ 3010 $ $ 1771 $ $ 1010 $ $ 158.2 $
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