Web of Science (Emerging Sources Citation Index), ISC

Document Type : Original Research Article

Authors

1 Department of Studies in Mathematics, University of Mysore, Manasagangotri Mysuru-570 006, India

2 Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysuru-570005, India

Abstract

In medical science, pharmacology, chemical, pharmaceutical properties of molecular structure are essential for drug preparation and design. These properties can be studied by using domination topological indices calculation. In this research work, we establish the topological properties of levodopa-carbidopa drug given to people with Parkinson’s disease by using the domination of topological indices and domination indices. We determine the φP-polynomial for the chemical structures of levodopa a

Graphical Abstract

Keywords

Main Subjects

###### ##### Full Text

Introduction

Chemical graph theory is one of the branches of mathematical chemistry. The importance of chemical graph theory lies in understanding and explaining the nature of the chemical composition, as it was used in organizing the current problem because it determines the arrangement of rules and laws according to a specific system and planning. The atoms are represented as the vertices and the chemical bonds between the atoms are the edges connecting these vertices. Topological indicators are molecular descriptors that describe the composition of chemical structures and help predict some of the chemical and physical properties of these structures. A set D sub set of V is said to be a dominating set of a graph G, if for any vertex v∈V −D there exists a vertex u∈D such that u and v are adjacent. For more details on domination in graphs, see [5,6,7,11,18-28]. A dominating set D={v1,v2,…,vr} is minimal if D −vi is not a dominating set [10]. A dominating set of G of minimum cardinality is said to be a minimum dominating set.

Definition 1.1. [12] For each vertex v∈V(G), the domination degree is denoted by dd(v) and defined as the number of minimal dominating sets of G which contains v.

Hanan Ahmed et al. [12] have introduced new topological indices called domination Zagreb indices which are based on the minimal dominating sets defined as:

Where dd(v) is the domination degree of the vertex v. The total number of minimal dominating sets of G is denoted as Tm(G) [12].

The forgotten domination, hyper domination and modified forgotten domination indices of graphs [13] are defined as:

Domination (D) and γ-Domination (γD) indices defined on E(G) can be written as:

For more information on topological indices and polynomial of graph see [1,2,3,4,9,8,15,16,17].

Results and discussion

Levodopa was developed over 30 years ago and is often considered the appropriate standard for Parkinson’s treatment. Levodopa works by crossing the blood-brain barrier, where it is converted to dopamine. The blood enzymes break down most of the levodopa before it reaches the brain. For this reason, levodopa is combined with an enzyme inhibitor called carbidopa. The addition of carbidopa prevents levodopa from being metabolized in the gastrointestinal tract and liver.

In this paper, we used the notations G=molecular graph of levodopa and H=molecular graph of carbidopa.

Lemma 2.1. Let G and H be the molecular graphs of levodopa and carbidopa, respectively. Then, Tm(G)=49, Tm(H)=96 and there are 6 and 10 minimum dominating sets in the molecular graphs of levodopa and carbidopa, respectively

Proof. Let G be the molecular graph of levodopa C9H11N O4. The minimal dominating sets of G are:

Note that among 49 minimal dominating sets, there are 6 minimum dominating sets. Similarly, if H be the molecular graph of carbidopa C10H14N2O4, one can get 96, 10 minimal and minimum dominating sets, respectively.

From Lemma 2.1, Definition 1.1 and Definition1.2, we get, if vi∈V(G) then:

Conclusion

We have studied and computed the properties of Carbidopa-Levodopa used for treatment Parkinson’s disease through domination and γ-Domination topological indices. First, we found φd polynomial and φγ polynomial and their respective 3D graphs (Figures 3 and 4).  Then we computed the domination and γ-Domination indices from these polynomials.

Acknowledgments

The authors extend their real appreciation to the reviewers for their insightful comments and technical suggestions to enhance quality of the article.

Orcid:

Hanan Ahmed: https://orcid.org/0000-0002-4008-4873

Ammar Alsinai: https://orcid.org/0000-0002-5221-0574

Soner Nandappa D.: https://orcid.org/0000-0003-1968-4097

-----------------------------------------------------------------------

How to cite this article: Swamy Javaraju, Hanan Ahmed*, Nandappa D. Soner, Ammar Alsinai. Domination topological properties of carbidopa-levodopa used for treatment Parkinson’s disease by using φp-polynomial. Eurasian Chemical Communications, 2021, 3(9), 614-621. Link: http://www.echemcom.com/article_134652.html

-----------------------------------------------------------------------

###### ##### References
[1] A. Alsinai, A. Alwardi, N.D. Soner, Int. J. Anal. Appl., 2021, 19, 1-19. [crossref], [Google Scholar], [Publisher]
[2] A. Alsinai, A. Alwardi, N.D. Soner, J. Dis. Math. Sci. Cryp., 2021, 24, 307-324. [crossref], [Google Scholar], [Publisher]
[3] A. Alsinai, A. Alwardi, N.D. Soner, Eurasian Chem. Commun., 2021, 3, 219-226. [crossref], [Google Scholar], [Publisher]
[4] M. Azari, A. Iranmanesh, MATCH Commun Math. Com. Chem., 2014, 71, 373-382. [Pdf], [Google Scholar], [Publisher]
[5] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. [Pdf], [Google Scholar], [Publisher]
[6] C. Berge, Theory of Graphs and its Applications, Methuen, London, 1962, 238-244. [Google Scholar], [Publisher]
[7] E.J. Cockayne, S.T. Hedetniemi, Networks, 1977, 7, 247-261. [crossref], [Google Scholar], [Publisher]
[8] T. Doslic, M. Ghorbani, M.A. Hosseinzadeh, Util. Math., 2011, 3, 1-10.
[9] T. Al-Fozan, P. Manuel, I. Rajasingh, R. Sundara Rajan, MATCH Commun Math. Com. Chem., 2014, 72, 339-353.  [Pdf], [Google Scholar], [Publisher]
[10] J.F. Couturier, R. Letourneur, M. Liedlo, Theor. Comput. Sci., 2015, 562, 634-642. [crossref], [Google Scholar], [Publisher]
[11] M. Imran, S.A. Bokhary, S. Manzoor, M.K. Siddiqui, Eurasian Chem. Communm., 2020, 2, 680-687. [crossref], [Google Scholar], [Publisher]
[12] H. Ahmed, A. Alwardi, M.R. Salestina, Int. J. Anal. Appl., 2021, 19, 47-64. [crossref], [Google Scholar], [Publisher]
[13] H. Ahmed, A. Alwardi, R. Salestina M., N.D. Soner, J. Dis. Math. Sci. Cryp., 2021, 24, 353-368. [crossref], [Google Scholar], [Publisher]
[14] H. Ahmed, A. Alwardi, R. Salestina M., S. Nandappa D.,  Biointerface Res. Appl. Chem., 2021, 11, 13290-13302. [crossref], [Google Scholar], [Publisher]
[15] H. Ahmed, M.R. Farahani, A. Alwardi, R. Salestina M., Eurasian Chem. Commun., 2021, 3, 210-218. [crossref], [Google Scholar], [Publisher]
[16] H. Ahmed, A. Alwardi, R. Salestina M., J. Dis. Math. Sci. Cryp., 2021, 24,   325-341. [crossref], [Google Scholar], [Publisher]
[17] H. Ahmed, A. Alwardi, R. Salestina M., Res. Rev.: Dis. Math. Str., 2020 7, 7-14. [crossref], [Google Scholar], [Publisher]
[18] O. Ore, Theory of Graphs, American Mathematical Society Colloquium Publications, 38 (American Mathematical Society, Providence, RI) 1962.
[19] D. Afzal, S. Hussain, M. Aldemir, M. Farahani, F. Afzal, Eurasian Chem. Commun., 2020, 2, 1117-1125. [crossref], [Google Scholar], [Publisher]
[20] F. Chaudhry, M. Ehsan, D. Afzal, M.R. Farahani, M. Cancan, E. Ediz, J. Dis. Math. Sci. Cryp., 2021, 24, 401-414. [crossref], [Google Scholar], [Publisher]
[21] F. Chaudhry, M. Ehsan, F. Afzal, M.R. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 146-153. [crossref], [Google Scholar], [Publisher]
[22] F. Chaudhry, M. Ehsan, F. Afzal, M.R. Farahani, M. Cancan, I. Ciftci, Eurasian Chem. Commun., 2021, 3, 103-109. [crossref], [Google Scholar], [Publisher]
[23] A.J.M. Khalaf, S. Hussain, D. Afzal, F. Afzal, A. Maqbool, J. Dis. Math. Sci. Cryp., 2020, 23, 1217-1237. [crossref], [Google Scholar], [Publisher]
[24] D.Y. Shin, S. Hussain, F. Afzal, C. Park, D. Afzal, M.R. Farahani, Frontier in Chemistry, 2021, 8, 613873-613877. [crossref], [Google Scholar], [Publisher]
[25] M. Cancan, S. Ediz, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 641-645. [crossref], [Google Scholar], [Publisher]     [26] A.Q. Baig, M. Naeem, W. Gao, J.B. Liu, Eurasian Chem. Commun., 2020, 2, 634-640. [crossref], [Google Scholar], [Publisher]
[27] F. Afzal, M.A. Razaq, D. Afzal, S. Hameed, Eurasian Chem. Commun., 2020, 2, 652-662. [crossref], [Google Scholar], [Publisher]
[28] M. Alaeiyan, C. Natarajan, G. Sathiamoorthy, M.R. Farahani, Eurasian Chem. Commun., 2020, 2, 646-651. [crossref], [Google Scholar], [Publisher]