July  2021, 14(7): 2487-2495. 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  July 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2487-2495. 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.

[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. 

[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. 

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[34]

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

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.

[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. 

[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. 

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[34]

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

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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