Document Type : Original Research Article
Authors
- Muhammad Ehsan ^{1}
- Sumera Sattar ^{2}
- Faryal Chaudhry ^{1}
- Farkhanda Afzal ^{} ^{} ^{3}
- Mohammad Reza Farahani ^{} ^{4}
- Murat Cancan ^{} ^{5}
^{1} Department of Mathematics and Statistics, The University of Lahore, Lahore, 54000, Pakistan
^{2} Department of Health and Physical Education, Lahore College for Women University, Lahore, Pakistan
^{3} MCS, National University of Science and Technology, Islamabad, Pakistan
^{4} Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran
^{5} Faculty of Education, Van Yuzuncu Yıl University, Van, Turkey
Abstract
The chemical graph theory is interrelated with the chemical structure of different compounds. This graph represents the molecule of the sub-stance. A chemical graph is rehabilitated into a real number by applying some mathematical tackles. This number can elaborate on the properties of the molecule. This number is called topological catalogs. Here, we find some topological catalogs via M-polynomial for the zigzag-edge coronoid graph.
Graphical Abstract
Keywords
Main Subjects
Introduction
Topological indices [1-10] in theoretical chemistry has drawn a great interest. The topological indices help us to recognize the different sorts of chemical substances. The topological index has a basic role that shows the chemical building to a mathematical number which is used to explain a molecule under testing.
Topological indices [11-18] are calculated from their definition; however, these are also calculated by using their M-polynomial. M-polynomial is also graph demonstrative mathematical object. By using M-polynomial, we work out various degree dependent topological invariant present in Table 2.
The M-polynomial introduced for a graph G is defined as [19]:
Where δ=min{d_{v}|v∈V (G)}, Δ=max{d_{v}|v∈V (G)} and m_{ij}(G) is the number of edges vu∈E(G) such that {d_{v};d_{u}}={i;j}.
Table 3 represents some well-known degree based topological directories to compute through M-Polynomial. M-polynomial of many graphs were introduced in the past [20-30]. In the current work, we add M-polynomials and topological indices [31-33] of ZC(g,h,q), Figure 1, of the zigzag-edge coronoid ZC(g,h,q), can be considered as a structure obtained by fusing three linear polyenes of length g; h and q, respectively.
M-Polynomial of zigzag-edge coronoid graph
Theorem 3.1. If Zigzag-edge coronoid is denoted by ZC(g,h,q), then for g,h,q≥3, its M-polynomial is
M[ZC(g,h,q),x,y]
=6x^{2}y^{2}+(8(g+h+q)-36)x^{2}y^{3}+2(g+h+q)x^{3}y^{3}
Proof. Let ZC(g,h,q) be a Zigzag-edge coronoid, then from Table 1 and figure 1 the Edge partition of ZC(g,h,q) is
E_{2;2}(ZC(g,h,q))={e=uv∈ZC(g,h,q): d_{u}=2; d_{v}=2} → |E_{2;2}ZC(g,h,q)|=6
E_{2;3}(ZC(g,h,q))={e=uv∈ZC(g,h,q): d_{u}=2; d_{v}=3} → |E_{2;3}ZC(g,h,q)|=(8(g+h+q)-36)
E_{3;3}ZC(g,h,q))={e=uv∈ZC(g,h,q): d_{u}=3; d_{v}=3} → |E_{3;3}ZC(g,h,q)|=2(g+h+q)
The following result obtained by applying the definition of M-polynomial
M[ZC(g,h,q),x,y]=|E_{2;2}|x^{2}y^{2}+|E_{2;3}|x^{2}y^{3}+|E_{3;3}|x^{3}y^{3}
=6x^{2}y^{2}+(8(g+h+q)-36)x^{2}y^{3}+2(g+h+q)x^{3}y^{3}.
The strategy of M-polynomial of ZC(g,h,q) is shown in Figure 2.
Topological indices of zigzag-edge coronoid
Theorem 4.1. Let ZC(g, h, q) be a Zigzag-edge coronoid
M[ZC(g,h,q),x,y]=6x^{2}y^{2}+(8(g+h+q)-36)x^{2}y^{3}+2(g+h+q)x^{3}y^{3 }
Proof. Let M[ZC(g,h,q),x,y]=6x^{2}y^{2}+(8(g+h+q)-36)x^{2}y^{3}+2(g+h+q)x^{3}y^{3}
Conclusion
In this study, we calculated the closed-form of M-polynomial for the graph Zigzag-edge coronoid, and then we derivative several degree-based topological indices as well, which supports shrinking the number of experiments. These topological indices can help characterize biological, chemical, and physical features of a molecule. So topological index has a fundamental role that represents the chemical structure of a molecule to a real number and is used to precise the molecule which is being tested. These outcomes are very supportive in accepting and forecasting the physico-chemical properties of these chemical structures. Distance-related graph indices for these imperative chemical graphs is still open to further research.
Acknowledgments
The authors extend their real appreciation to the reviewers for their insightful comments and technical suggestions to enhance quality of the article.
Orcid:
Farkhanda Afzal: https://orcid.org/0000-0001-5396-7598
Mohammad Reza Farahani: https://orcid.org/0000-0003-2969-4280
Murat Cancan: https://orcid.org/0000-0002-8606-2274
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How to cite this article: Muhammad Ehsan, Sumera Sattar, Faryal Chaudhry, Farkhanda Afzal*, Mohammad Reza Farahani, Murat Cancan. Topological analysis of zigzag-edge coronoid graph by using M-polynomial. Eurasian Chemical Communications, 2021, 3(9), 590-597. Link: http://www.echemcom.com/article_134628.html
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Copyright © 2021 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY) license (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.