Abstract
Dendrimers, also known as dendritic polymers, have various applications due to their unique properties, such as their monodisperse structure and their ability to be synthesized with precise control over their size, shape, and surface functionality. Dendrimers are used in drug delivery systems to improve drug solubility, bioavailability, and targeting. They can carry drugs to specific sites, such as cancer cells, and release them in a controlled manner, reducing side effects. Dendrimers can be used as gene delivery vehicles to deliver genetic material to cells in a controlled and targeted manner. Mathematical chemistry is useful to model chemical reactions and predict the behavior of chemical systems. It provides a quantitative understanding of chemical phenomena, which can aid in the design of new molecules and materials. It is used to develop molecular descriptors, which are mathematical representations of molecular structures that can be used to quantify the properties of molecules. These descriptors can be useful in structure–activity relationship studies to predict the biological activity of compounds. The topological descriptors are parameters of any molecular structure that gives a mathematical formula to model such molecular structures. In the current study, our concern is to calculate some useful topological indices for three kinds of dendrimer networks and derive closed mathematical formulas for them. The comparisons of these calculated topological indices are also investigated. Our obtained results will be helpful in investigating QSPRs/QSARs of such molecules in many fields of science, such as chemistry, physics and biochemistry.
Graphical abstract
The dendrimer structure (left). From first (G0) to third (G3) generation, the dendrimer's increasing generations are shown schematically (right).
Similar content being viewed by others
Availability of data and materials
All data generated or analyzed during this study are included in this article.
References
H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1), 17–20 (1947)
A. Ahmad, O.B.S. Al-Mushayt, M. Bača, On edge irregularity strength of graphs. Appl. Math. Comput. Biol. 243, 607–610 (2014)
A. Ahmad et al., Computing the degree based topological indices of line graph of benzene ring embedded in P-type-surface in 2D network. J. Inf. Optim. Sci. 40(7), 1511–1528 (2019)
A.A. Khabyah et al., Minimum zagreb eccentricity indices of two-mode network with applications in boiling point and benzenoid hydrocarbons. Mathematics 10(9), 1393 (2022)
A. Ullah et al., Network-based modeling of the molecular topology of fuchsine acid dye with respect to some irregular molecular descriptors. J. Chem. 2022, 8131276 (2022)
A. Ahmad et al., Polynomials of degree-based indices for swapped networks modeled by optical transpose interconnection system. IEEE Access 8, 214293–214299 (2020)
A. Ahmad, Computation of certain topological properties of para-line graph of honeycomb networks and graphene. Discrete Math. Algorithms Appl. 9(05), 1750064 (2017)
S. Zaman et al., The kemeny’s constant and spanning trees of hexagonal ring network. Comput. Mater. Contin. 73, 6347–6365 (2022)
S. Zaman et al., Structural analysis and topological characterization of sudoku nanosheet. J. Math. 2022, 5915740 (2022)
A. Ullah et al., Zagreb connection topological descriptors and structural property of the triangular chain structures. Phys. Scr. 98(2), 025009 (2023)
A. Ullah, A. Zeb, S. Zaman, A new perspective on the modeling and topological characterization of H-Naphtalenic nanosheets with applications. J. Mol. Model. 28(8), 211 (2022)
S. Zaman et al., On the topological descriptors and structural analysis of cerium oxide nanostructures. Chem. Pap. 77, 2917–2922 (2023)
X. Yu et al., Matrix analysis of hexagonal model and its applications in global mean-first-passage time of random walks. IEEE Access 11, 10045–10052 (2023)
A. Ullah et al., Computational and comparative aspects of two carbon nanosheets with respect to some novel topological indices. Ain Shams Eng. J. 13(4), 101672 (2022)
S. Zaman, A. Ullah, Kemeny’s constant and global mean first passage time of random walks on octagonal cell network. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9046
M.O. Albertson, The irregularity of a graph. ARS Comb. 46, 219–225 (1997)
D. Vukičević, A. Graovac, Valence connectivity versus Randić, Zagreb and modified Zagreb index: a linear algorithm to check discriminative properties of indices in acyclic molecular graphs. Croat. Chem. Acta 77(3), 501–508 (2004)
I. Gutman, Topological indices and irregularity measures. J. Bull. 8, 469–475 (2018)
D. Dimitrov, S. Brandt, H. Abdo, The total irregularity of a graph. Discrete Math. Theor. Comput. Sci. (2014). https://doi.org/10.46298/dmtcs.1263
X. Li, I. Gutman, Mathematical Aspects of Randic-Type Molecular Structure Descriptors. Mathematical Chemistry Monographs No. 1, Kragujevac, vol. 1 (Faculty of Science, University of Kragujevac, Kragujevac, 2006)
T. Réti et al., Graph irregularity indices used as molecular descriptors in QSPR studies. MATCH Commun. Math. Comput. Chem. 79, 509–524 (2018)
A. Avdullahu, S. Filipovski, On Certain Topological Indices of Graphs (2022). arXiv preprint https://arxiv.org/abs/2210.12981
G. Hong et al., Degree-based topological invariants of metal-organic networks. IEEE Access 8, 68288–68300 (2020)
D. Kovačević, A. Graovac, Valence connectivities versus Randić, Zagreb and modified Zagreb index: a linear algorithm to check discriminative properties of indices in acyclic molecular graphs. Croa. Chem. Acta 77, 501–508 (2004)
A.S. Abd-El-Aziz, C. Agatemor, Emerging opportunities in the biomedical applications of dendrimers. J. Inorg. Organomet. Polym. Mater. 28, 369–382 (2018)
L. Palmerston Mendes, J. Pan, V.P. Torchilin, Dendrimers as nanocarriers for nucleic acid and drug delivery in cancer therapy. Molecules 22(9), 1401 (2017)
Funding
No funding is available for this study.
Author information
Authors and Affiliations
Contributions
All the authors have equally contributed to this manuscript in all stages, from conceptualization to the write-up of final draft.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no any conflict of interest.
Ethical approval
Not applicable.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zaman, S., Ullah, A. & Shafaqat, A. Structural modeling and topological characterization of three kinds of dendrimer networks. Eur. Phys. J. E 46, 36 (2023). https://doi.org/10.1140/epje/s10189-023-00297-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/s10189-023-00297-4