Document Type : Original Research Article
Authors
- Farkhanda Afzal ^{} ^{} ^{1}
- Ammar Alsinai ^{} ^{2}
- Mohammad Zeeshan ^{3}
- Deeba Afzal ^{} ^{3}
- Faryal Chaudhry ^{3}
- Murat Cancan ^{} ^{4}
^{1} MCS, National University of Sciences and Technology, Islamabad, Pakistan
^{2} Department of Studies in Mathematics University of Mysore, Manasagangotri, India
^{3} Department of Mathematics and Statistics, The University of Lahore, Lahore, 54000, Pakistan
^{4} Faculty of Education, Van Yuzuncu Yıl University, Zeve Campus, Tuşba, 65080, Van, Turkey
Abstract
In this object, we present some new formulas of the reduced reciprocal Randić index, the arithmetic geometric _{1} index, the SK, SK_{1}, SK_{2} indices, first Zagreb index, the general sum-connectivity index, the SCI index and the forgotten index. They were utilized for new degree-based topological indices via M-polynomial. We retrieved these topological indices for H-Naphtalenic nanotubes.
Graphical Abstract
Keywords
Introduction
Chemical graph theory is a branch of mathematics, which combines chemistry and graph theory. Graph theory is a mathematical imitation of molecules to construct insight into the properties of those chemical mixtures. Each carbon atom forms four bonds in a stable organic compound, such as ethane, ethene (ethylene), and ethyne (acetylene). The carbon atom in ethene forms four single bonds, each of the three hydrogen atoms and one to the neighboring carbon atoms of each hydrogen atom has one chemical bond.
The topological indices assist in the physical features, chemical and bio-logical reactivity. The topological index is surveying the principal part that shows each molecular structure of the real number, implemented as a descriptor of the molecule under checking. This graphical representation of topological indices shows the dependency of a confirmed index on the structure.
A nanotube is chiefly sheets build up into a tube. Nanotubes concern different graph-theoretic frameworks in graph theory, like polynomials and topological indices. H-Naphtalenic is rising in the same way by a sheet cover with squares, hexagons and octagons. The topological indices are also computed via M-polynomial. Several works have already been done in this area [1,2,3-6,22,23].
The M-polynomial introduced by Emeric Deutsch and Sandi Klavžar [7] is defined for a graph G as the first topological index invented by wiener [8]. Topological indices are used in the development, quantitative activity and also other properties of molecules correlate in chemical structure. The connection between atoms is shown as topological indices of different chemical compound such as boiling point, the heat of formation, heat of evaporation, surface tension, vapor pressure and etc. Now we define some topological indices.
In 2015 [9], I. Gutman ed ul., invented a reduced reciprocal index. The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (more precisely, a topological index), which is helpful for the divine level of enthalpy creation and the usual boiling point of isomeric octanes.
In 2016 [10]. V. Shegehalli and R. Kanabur are used arithmetic geometric index.
The SK, SK_{1}, SK_{2} indices are defined as:
In 2016 [10] Shegehalli and Kanabur used SK, SK_{1}, SK_{2} indices.
Where
H-naphtalenic nanotube
Carbon nanotubes were initially used in 1991, and one-dimensional material appeared. Nanotubes are created by rolling up sheets into a tube. The nanotubes are largely studied in graph theory such as polynomials and topological indices. H-Naphtalenic nanotubes are formed in same way with sheets covered by squares, hexagons and octagons.
Let HNnt_{mn} be a H-Naphtalenic nanotube then for m; n≥1, M-polynomial of HNnt_{mn} is given as:
M(HNnt_{mn};x;y)=8nx^{2}y^{3}+5n(3m-2)x^{3}y^{3}
Topological indices of H-Naphtalenic nanotube via M-polynomial
Theorem 3.1. Let HNnt be a H-Naphtalenic nanotube then for m; n≤ 1, M-polynomial is given as:
M(HNnt_{mn};x;y)=8nx^{2}y^{3}+5n(3m-2)x^{3}y^{3:}
Proof. Let HNnt be a H-Naphtalenic nanotube. The M-polynomial is given as:
M(HNnt_{mn};x;y)=8nx^{2}y^{3}+5n(3m-2)x^{3}y^{3}
Conclusion
The induction of new closed formulas was to count the topological indices via M-polynomial. In this study, we determined M-polynomials of H-Naphtalenic (HNnt_{mn}). These topological indices are useful for the augury plan of physical features, biological activities, and chemical reactivates of the substance.
Acknowledgments
The authors would like to thank the reviewers for their helpful suggestions and comments.
Orcid:
Farkhanda Afzal: https://orcid.org/0000-0001-5396-7598
Ammar Alsinai: https://orcid.org/0000-0002-5221-0574
Deeba Afzal: https://orcid.org/0000-0001-5268-7260
Murat Cancan: https://orcid.org/0000-0002-8606-2274
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How to cite this article: Farkhanda Afzal*, Ammar Alsinai, Mohammad Zeeshan, Deeba Afzal, Faryal Chaudhry, Murat Cancan. Some new degree based topological indices of h-naphtalenic graph via M-polynomial approach. Eurasian Chemical Communications, 2021, 3(11), 800-805. Link: http://www.echemcom.com/article_137562.html
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