Web of Science (Emerging Sources Citation Index), Scopus, ISC

Document Type : Original Research Article

Authors

1 Department of Mathematices, Maharanis Science College for Women, Mysore 570005, India

2 bDepartment of Mathematics, Yuvarajas College, University of Mysore, Mysore 570005, India

Abstract

In this work, Mhr- polynomial of oxybenzone, menthyl anthranilate, benzophenone-4, and dimethylamino hydroxybenzoyl hexa benzoate were established. The degree-based topological indices HDR version of Modified Zagreb topological index (HDRM*), HDR version of Modified forgotten topological index (HDRF*), and HDR version of hyper Zagreb index (HDRHM*) were obtained. Accordingly, by using the derivative of Mhr- polynomial of sunscreens, the HDRM*, HDRF*, and HDRHM* topological indices of sunscreens were found.

Graphical Abstract

Topological properties of sunscreens using mhr-polynomial of graph

Keywords

Main Subjects

Introduction

A molecular graph is an undirected graph. It is denoted by G=(V, E), which shows the general properties of the molecular compound, where the vertices of the shovel |V| represent the number of atoms. In contrast, the edges E represent the juxtaposition relationship between those vertices that represent the atoms of the molecular compound [1]. In a molecular graph, the vertices represent atoms, and the edges represent chemical bonds, and they correspond to them. Suppose the G graph represents a chemical compound that contains a set of vertices V(G), and a set of edges is E(G). In that case, the degree of the vertex can be defined as the number of neighbors of the vertex in and is represented define by dG(v), which is the total number of edges associated with v [2]. Topological indices are essential relationships in modeling quantitative relationships for the structure and activity of chemical graphs [3]; for more didcestion about topological indices [7-30].

Topological indicators are fixed values in a graph by which the real values are assigned, taking the graph as a consistent median and giving similar graphs the same value. Wiener index gave the first topological index [2].

In 1947, for studying boiling points of alkanes, one of the topological indices invented at the preliminary level is the so-called Zagreb index, first provided by [4,5]. They investigated how an electron whole energy relied on the shape of molecules and became discussed. The primary Zagreb indices 

DAlsinai A, at al, [1] define (1):

HDR of Zagreb index version modified of Hyper HDR Zagreb indices modified forgotten topological index of graph G by:

 

Materials and methods

Our main results including comuting Mhr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, the degree-based topological indices HDRM*, HDRF*, and HDRHM* obtained by using the derivative of Mhr- polynomial of sunscreens the HDRM*,  HDRF*, and HDRHM* topological indices of sunscreens, are found.

Main results

This section includes two subsections computation Mhr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, and computation topological indices of sunscreens (Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate).

Evalute the - polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate

In this subsection, we compute the Polynomial of the sunscreens structures and we represent Mhr (3D) graphical using MATLAB.

Oxybenzone

Let  be a graph of oxybenzone, as shown in Figure 1. Then the graph 𝐺 has 17 vertices and 17 edges.

Theorem 1. Let 𝐺 be a graph of oxybenzone. Then

Proof. Let G a graph of Oxybenzone as in Figure 1. Then by using Table 2 and Definition2, we have:

Menthyl anthranilate

Let G be a graph of Menthyl Anthranilate, as shown in Figure 3. The graph 𝐺 contain 18 vertices and 19 edges.

Theorem 2. Let 𝐺 be a graph of menthyl anthranilate. Then

Proof. Let G be a graph of Menthyl Anthranilate as in Figure3. Then by using Table 3 and Definition 2, we have

Benzophenone-4

Let 𝐺 be a graph of Benzophenone-4, as shown in Figure 5. Then the graph 𝐺 contain 21 points and 22 edges and

Theorem 3. Let 𝐺 be the graph of benzophenone-4. Then

Proof. Let G be the graph of Benzophenone-4, as seen in Figure 5. Then by using Table 4 and Definition 2, we have:

Dimethylamino hydroxybenzoyl hexa benzoate

Let 𝐺 be the graph of Dimethylamino hydroxybenzoyl hexa benzoate, as shown in Figure 7. The graph 𝐺 contains 21 points and 22 edges.

Theorem 4. Let 𝐺be a moleculargraph of Dimethylamino hydroxybenzoyl hexa benzoate, then

Proof. Let Ga molecular graph of Dimethylamino hydroxybenzoyl hexa benzoate, as seen in Figure 7. Then by using Table 5 and Definition 2, we have

Compution HDR topological indices of sunscreens (Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate)

In this subsection, we computed HDR indices of Modified Zagreb topological index (HDRM*), (HDRF*), and (HDRHM*)   by using the derivative of Mhr- polynomial of sunscreens

Theorem 5. Let 𝐺 be the graph of Oxybenzone, and 

Using Table 1, we find the outcome.■

Theorem 6. Let 𝐺 be a molecular graph of Menthyl Anthranilate, and 

Using Table 1, we find the outcome .■

Theorem 7. Let 𝐺 be a molecular graph of Benzophenone-4, and 

Using Table 1, we find the outcome   .■

Theorem 8. Let 𝐺 be a molecular graph of Dimethylamino hydroxybenzoyl hexa benzoate, and 

Conclusion

We found Mhr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, the degree-based topological indices  HDRM*, HDRF*, HDRHM*  obtained. By using the derivative of Mhr- polynomial of sunscreens the HDRM*, HDRF*, and HDRHM* topological indices of sunscreens, were found. The results of the HDRM*, HDRF*, and HDRHM* of sunscreens are better than the leap indices and Zagreb Indices.

Orcid:

Jyoth. MJ: https://orcid.org/0000-0002-4682-4713

Shivashankara: https://orcid.org/0000-0001-9473-0474

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How to cite this article: Jyothi. MJ*, K. Shivashankara. Topological properties of sunscreens using mhr-polynomial of graph. Eurasian Chemical Communications, 2022, 4(5), 402-410. Link: http://www.echemcom.com/article_145641.html

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Copyright © 2022 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.

[1] A. Alsinai, H. Ahmed, A. Alwardi, S. Nandappa, D., Biointerface Res. Appl. Chem., 2021, 12, 7214-7225. [Crossref], [Google Scholar], [Publisher]
[2] S.N. Ilemo, D barth, D. David, F. Quessette, M.A. Weisser, PLoS ONE, 2019, 14, 1-25. [Crossref], [Google Scholar], [Publisher]
[3] S. Mondal, N. De, A. pal, ACTA CHEMICA IASI 2019, 27, 31-46. [Crossref], [Google Scholar], [Publisher]
[4] A.R. Ashrafi, M. Saheli, M. Ghorbani, J. Comput. Appl. Math, 2011, 235, 4561-4566. [Crossref], [Google Scholar], [Publisher]
[5] I. Gutman, B. Ru´ci´c, N. Trinajesti´c, C.F. Wilcox, J. Chem. Phys., 1975, 62, 3399. [Crossref], [Google Scholar], [Publisher]
[6] I. Gutman, N. Trinajesti´c, Chem. Phys, 1972, 17, 535-538. [Crossref], [Google Scholar], [Publisher]
[7] I Gutman, K.C. Das, MATCH Commun. Math. Comput. Chem., 2004, 50, 83-92. [Pdf], [Google Scholar], [Publisher]
[8] I. Gutman, A.M. Naji, N.D. Soner, Commun. Comb. Optim., 2017, 2, 99-117. [Crossref], [Google Scholar], [Publisher]
[9] V.R. Kulli, International Journal of Current Researcher in Life Scienes, 2018, 7, 2783-2791. [Pdf], [Google Scholar], [Publisher]
[10] B. Furtula, I. Gutman, J. Math. Chem., 2015, 53, 1184-1190. [Crossref], [Google Scholar], [Publisher]
[11] A. Bharali, A. Doley, J. Buragohain, Mathematical Computation in Combinatorics and Graph Theory, 2020, 39, 1019-1032. [Crossref], [Google Scholar], [Publisher]
[12] A. Alsinai, A. Alwardi, N.D. Soner, Int. J. Anal. Appl., 2020, 19, 1-19. [Crossref], [Google Scholar], [Publisher]
[13] K.C. Das, K. Xu, I. Gutman, MATCH Commun. Math. Comput. Chem., 2013, 70, 301-314. [Pdf], [Google Scholar], [Publisher]
[14] N. Chidambaram, S. Mohandoss, X. Yu, X. Zhang, AIMS mathematices, 2020, 6, 6521-6536. [Crossref], [Google Scholar], [Publisher]
[15] A. Alsinai, A. Alwardi, N.D. Soner, Eurasian Chem. Commun., 2021, 3, 219-226. [Crossref], [Google Scholar], [Publisher]
[16] A. Alsinai, A. Alwardi, N.D. Soner, J. Discrete Math. Sci. Cryptogr., 2021, 24, 307-324. [Crossref], [Google Scholar], [Publisher]
[17] A. Alsinai, A. Alwardi, N.D. Soner, Proc. Jangjeon Math. Soc., 2021, 24, 375–388. [Crossref], [Google Scholar], [Publisher]
[18] F. Afzal, A. Alsinai, S. Hussain, D. Afzal, F. Chaudhry, M. Cancan, J. Discrete Math. Sci. Cryptogr., 2021, 24, 1-11. [Crossref], [Google Scholar], [Publisher]
[19] V.R. Kulli, Int. J. Eng. Sci. Technol., 2020, 9, 73-84. [Crossref], [Google Scholar], [Publisher]
[20] I. Matečića, P.R.X. Šikanjićb, A.P. Lewis, J. Herit. Tour., 2021, 16, 450-468. [Crossref], [Google Scholar], [Publisher]
[21] F. Afzal, D. Afzal, A.Q. Baig, M.R. Farahani, M. Cancan, E. Ediz, Eurasian Chem. Commun., 2020, 2, 1183-1187. [Crossref], [Google Scholar], [Publisher]
[22] E. Schrezenmeier, T. Dörner, Nat. Rev. Rheumatol., 2020, 16, 155-166. [Crossref], [Google Scholar], [Publisher]
[23] S. Hussain, A. Alsinai, D. Afzal, A. Maqbool, F. Afzal, M. Cancan, Chem. Methodol., 2021, 5, 485-497. [Crossref], [Google Scholar], [Publisher]
[24] S. Mondal, N. De, A. Pal, Polycycl. Aromat. Compd., 2020, 1-15. [Crossref], [Google Scholar], [Publisher]
[25] M. Wang, R. Cao, L. Zhang, X. Yang, J. Liu, M. Xu, Z. Shi, Z. Hu, W. Zhong, G. Xiao, Cell Research, 2020, 30, 269-271. [Crossref], [Google Scholar], [Publisher]
[26] S. Mondal, N. De, A. Pal, Konuralp J. Math., 2020, 8, 97-105. [Pdf], [Google Scholar], [Publisher]
[27] J.B. Liu, J. Zhao, H. He, Z. Shao, J. Stat. Phys., 2019, 177, 1131-1147. [Crossref], [Google Scholar], [Publisher]
[28] F. Afzal, S. Hussain, D. Afzal, S. Razaq, Journal of Information and Optimization Sciences, 2020, 41, 1061-1076. [Crossref], [Google Scholar], [Publisher]
[29] A. Alsinai, A. Alwardi, H. Ahmed, N.D. Soner, J. Prime Res. Math., 2021. 17, 73-78. [Pdf], [Google Scholar], [Publisher]
[30] S. Javaraju, H. Ahmed, A. Alsinai, N.D. Soner, Eurasian Chem. Commun., 2021, 3, 614-621. [Crossref], [Google Scholar], [Publisher]