# The Study about Relationship of Direct Form of Topological Indices via M-Polynomial and Computational Analysis of Dexamethasone

Authors: Sabir Hussain; Farkhanda Afzal; Deeba Afzal; Dhana Kumari Thapa
Journal: Journal of Chemistry
 (2022)

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## Abstract

Nowadays, all scientists are in the race to find a cure for COVID-19. This is a viral disease that is due to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Different antiviral drugs are under investigation. Dexamethasone is being used for the treatment of COVID-19. In this article, the chemical structure of dexamethasone is explored using computational techniques such as topological indices. Proofs of a few closed formulas that describe the degree-dependent topological indices calculated from the M-polynomial of the graphs are also given in this article.


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## Body

## 1. Introduction

Epidemics such as flu, cholera, and plague disturb the life of the people of the world for centuries. COVID-19 began in China on December 19 and has since claimed many lives and is responsible for the largest global recession. This is due to the virus named severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [1, 2]. The spread of this virus can be reduced by using personal protective equipment and physical distance. The search for a cure to prevent COVID-19 is an important need of today.Different techniques such as antiviral drugs, vaccines, and plasma therapy are under progress to control this viral disease [3]. So far, no effective medicine has been developed to treat patients with COVID-19. Several clinical trials have been conducted to study the effect of different drugs on patients [4]. The results of the dexamethasone on patients with COVID-19 are encouraging [5, 6]. Dexamethasone is a corticosteroid that is used for its immunosuppressant and anti-inflammatory properties. The chemical formula of dexamethasone is C22H29FO5 and also known as 9α-fluoro-11β, 17α, 21-trihydroxy-16α-methylpregna-1,4-diene-3,20-dione. Dexamethasone is white to off-white crystalline powder. It has a slightly bitter taste and is odorless. 3D and 2D structures of dexamethasone are shown in Figure 1.Figure 1
Molecular structure of dexamethasone: (a) 3D structure; (b) 2D structure.
(a)(b)A graph is represented withGVG,EG where VG and EG have represented the vertex set and edge set, respectively. The number of edges that meet at a vertex x is called the degree of a vertex dx. The graphical form of the structure of the chemical compound is known as a chemical graph. In a chemical graph, atoms behave as vertex and bonds as edges. This graph is a valuable source of the physical and chemical properties of the substance. Computational analysis of the chemical graph has been studied in chemical graph theory. C-H bond does not have a serious effect on the characteristics of the chemical species. So, during the computational analysis, we ignore it.During computational analysis, a chemical graph is converted into a real number called topological index that can reflect the different physical, chemical, and biological features of the chemical compounds [7]. The study of the topological index is started by the formulation of the Wiener index [8]. Different topological applications have been found in [9–14]. For a graph G, a degree-dependent topological index is defined as(1)IG=∑e=xy∈EGfdx,dy.By counting edges that have the same end-degrees in the chemical graph, then we can rewrite equation (1) as(2)IG=∑j≤kmjkfj,k,where the relation dx,dy=j,k satisfied and mjk is the total count of edges xy of the graph G. In 2015, Gutman et al. [15] formulated a reduced reciprocal Randic’ index, which can be defined as(3)RRRG=∑xy∈EGdx−1dy−1.In 2016, Shegehalli and Kanabur presented the first arithmetic-geometric index [16] and defined as(4)AG1G=∑xy∈EGdx+dy2dx⋅dy.Shigehalli and Kanabur also introduced the three new indices [17] defined as(5)SKG=∑xy∈EGdx+dy2,SK1G=∑xy∈EGdx⋅dy2,SK2G=∑xy∈EGdx+dy22.Miličcević et al. presented a first Zagreb index in terms of edge degree [18] defined as(6)EM1G=∑xy∈EGdx+dy−22.Du et al. [19] formulated the general sum-connectivity index which can be described as(7)SCIG=∑xy∈EG1dx+dy,SCIλG=∑xy∈EGdx+dyλ.Ranjini et al. [20] redefined the Zagreb indices as(8)ReZG3G=∑xy∈EGdxdydx+dy.These topological indices are either calculated directly by their formula or by using the graph polynomials such as M-polynomial. Deutsch and Klawzar [21] define the M-polynomial as(9)MGu,v=∑ψ≤j≤k≤Ψmjkujvk.Here,ψ=mindx|x∈VG, Ψ=maxdx|x∈VG Numerous graphs have been studied in the past through M-polynomial and topological indices [22–24]. Some operators, which are used further, are defined as(10)DuMGu,v=u∂∂uMGu,v,DvMGu,v=v∂∂vMGu,v,Du1/2MGu,v=u∂∂uMGu,v⋅MGu,v,Dv1/2MGu,v=v∂∂yMGu,v⋅MGu,v,Su1/2MGu,v=∫0uMGt,vtdt⋅MGu,v,Sv1/2MGu,v=∫0vMGu,ttdt⋅MGu,v,QuαMGu,v=uαMGu,v,JMGu,v=MGu,u,QvαMGu,v=vαMGu,v.Shin et al. [25] presented the closed form of topological indices via M-polynomial, but they are not provided the relationship between them. In the following section, we provide this relationship.


## 2. Main Results

Theorem 1.
LetMGu,v be the M-polynomial for the graph G, then the reduced reciprocal index is calculated as RRRG=Du1/2Dv1/2Qu−1Qv−1MGu,v|u=v=1.Proof.
By taking(11)Du1/2Dv1/2Qu−1Qv−1MGu,v=Du1/2Dv1/2Qu−1Qv−1∑ψ≤j≤k≤Ψmjkujvk=Du1/2Dv1/2Qu−1∑ψ≤j≤k≤Ψmjkujvk−1=Du1/2Dv1/2∑ψ≤j≤k≤Ψmjkuj−1vk−1=Du1/2∑ψ≤j≤k≤Ψk−1mjkuj−1vk−1=∑ψ≤j≤k≤Ψj−1k−1mjkuj−1vk−1,Du1/2Dv1/2Qu−1Qv−1MGu,v|u=v=1=∑ψ≤j≤k≤Ψj−1k−1mjk,Du1/2Dv1/2Qu−1Qv−1MGu,v|u=v=1=∑xy∈EGdx−1dy−1.
Hence,RRRG=Du1/2Dv1/2Qu−1Qv−1MGu,v|u=v=1.Theorem 2.
IfMGu,v is the M-polynomial of the graph G, then the first arithmetic-geometric index is given by AG1G=1/2DuJSu1/2Sv1/2MGu,v|u=1.Proof.
By taking(12)12DuJSu1/2Sv1/2MGu,v=12DuJSu1/2Sv1/2∑ψ≤j≤k≤Ψmjkujvk=12DuJSu1/2∑ψ≤j≤k≤Ψ1kmjkujvk=12DuJ∑ψ≤j≤k≤Ψ1jkmjkujvk=12Du∑ψ≤j≤k≤Ψ1jkmjkuj+k=12∑ψ≤j≤k≤Ψj+kjkmjkuj+k=∑ψ≤j≤k≤Ψj+k2jkmjkuj+k,12DuJSu1/2Sv1/2MGu,v|u=1=∑ψ≤j≤k≤Ψj+k2jkmjk,12DuJSu1/2Sv1/2MGu,v|u=1=∑xy∈EGdx+dy2dx⋅dy.
Hence,AG1G=1/2DuJSu1/2Sv1/2MGu,v|u=1.Theorem 3.
IfMGu,v is represented the M-polynomial of the graph G, then SKG=Du+Dv/2MGu,v|u=v=1.Proof.
By taking(13)12Du+DvMGu,v=12Du+Dv∑ψ≤j≤k≤Ψmjkujvk=12Du∑ψ≤j≤k≤Ψmjkujvk+Dv∑ψ≤j≤k≤Ψmjkujvk=12∑ψ≤j≤k≤Ψjmjkujvk+∑ψ≤j≤k≤Ψkmjkujvk=12∑ψ≤j≤k≤Ψj+kmjkujvk=∑ψ≤j≤k≤Ψj+k2mjkujvk,Du+Dv2MGu,v|u=v=1==∑ψ≤j≤k≤Ψj+k2mjk,Du+Dv2MGu,v|u=v=1=∑xy∈EGdx+dy2.
Hence,SKG=Du+Dv/2MGu,v|u=v=1.Theorem 4.
IfMGu,v is the M-polynomial of the graph G, then SK1G=1/2DuDvMGu,v|u=v=1.Proof.
By taking(14)12DuDvMGu,v=12DuDv∑ψ≤j≤k≤Ψmjkujvk=12Du∑ψ≤j≤k≤Ψkmjkujvk=12∑ψ≤j≤k≤Ψjkmjkujvk=∑ψ≤j≤k≤Ψjk2mjkujvk,12DuDvMGu,v|u=v=1=∑ψ≤j≤k≤Ψjk2mjk,12DuDvMGu,v|u=v=1=∑xy∈EGdx⋅dy2.
Hence,SK1G=1/2DuDvMGu,v|u=v=1.Theorem 5.
IfMGu,v is the M-polynomial of the graph G, then SK2G=1/4Du2JMGu,v|u=v=1.Proof.
By taking(15)14Du2JMGu,v=14Du2J∑ψ≤j≤k≤Ψmjkujvk=14Du2∑ψ≤j≤k≤Ψmjkuj+k=14∑ψ≤j≤k≤Ψj+k2mjkuj+k=∑ψ≤j≤k≤Ψj+k24mjkuj+k,14Du2JMGu,v|u=v=1=∑ψ≤j≤k≤Ψj+k24,14Du2JMGu,v|u=v=1=∑xy∈EGdx+dy22.
Hence,SK2G=1/4Du2JMGu,v|u=v=1.Theorem 6.
LetMGu,v be the M-polynomial for the graph G, then EM1G=Du2Qu−2JMGu,v|u=v=1.Proof.
By taking(16)Du2Qu−2JMGu,v=Du2Qu−2J∑ψ≤j≤k≤Ψmjkujvk=Du2Qu−2∑ψ≤j≤k≤Ψmjkuj+k=Du2∑ψ≤j≤k≤Ψmjkuj+k−2=∑ψ≤j≤k≤Ψj+k−22mjkuj+k−2,Du2Qu−2JMGu,v|u=v=1=∑ψ≤j≤k≤Ψj+k−22mjk,Du2Qu−2JMGu,v|u=v=1=∑xy∈EGdx+dy−22.
Hence,EM1G=Du2Qu−2JMGu,v|u=v=1.Theorem 7.
LetMGu,v be the M-polynomial for the graph G, then the sum-connectivity index is given by SCIG=Su1/2JMGu,v|u=v=1.Proof.
By taking(17)Su1/2JMGu,v=Su1/2J∑ψ≤j≤k≤Ψmjkujvk=Su1/2∑ψ≤j≤k≤Ψmjkuj+k=∑ψ≤j≤k≤Ψ1j+kmjkuj+k,Su1/2JMGu,v|u=v=1=∑ψ≤j≤k≤Ψ1j+kmjkuj+k,Su1/2JMGu,v|u=v=1=∑xy∈EG1dx+dy.
Hence,SCIG=Su1/2JMGu,v|u=v=1.Theorem 8.
IFMGu,v is the M-polynomial of the graph G, then the general sum-connectivity index is given by SCIλG=DuλJMGu,v|u=v=1.Proof.
By taking(18)DuλJMGu,v=DuλJ∑ψ≤j≤k≤Ψmjkujvk=Duλ∑ψ≤j≤k≤Ψmjkuj+k=∑ψ≤j≤k≤Ψj+kλmjkuj+k,DuλJMGu,v|u=v=1=∑ψ≤j≤k≤Ψj+kλmjk,DuλJMGu,v|u=v=1=∑xy∈EGdx+dyλ.
Hence,SCIλG=DuλJMGu,v|u=v=1.Theorem 9.
IfMGu,v is the M-polynomial of the graph G, then the redefined third Zagreb index is also calculated as ReZG3G=DuDvDu+DvMGu,v|u=v=1.Proof.
By taking(19)DuDvDu+DvMGu,v=DuDvDu+Dv∑ψ≤j≤k≤Ψmjkujvk=DuDv∑ψ≤j≤k≤Ψj+kmjkujvk=Du∑ψ≤j≤k≤Ψkj+kmjkujvk=∑ψ≤j≤k≤Ψjkj+kmjkujvk,DuDvDu+DvMGu,v|u=v=1=∑ψ≤j≤k≤Ψjkj+kmjk,DuDvDu+DvMGu,v|u=v=1=∑xy∈EGdxdydx+dy.
Hence,ReZG1G=DuDvDu+DvMGu,v|u=v=1.


## 3. M-Polynomial of Dexamethasone

The chemical graph of dexamethasoneDM is shown in Figure 2, in which green, red, pink, and brown dots represent the vertices of degrees 1, 2, 3, and 4, respectively; cyan, orange, yellow, purple, blue, black, gray, violet, and magenta edges represent the edges having the degree of end-vertices 1,2, 1,3, 1,4, 2,2, 2,3, 2,4, 3,3, 3,4, and 4,4, respectively. In this present work, we extract some topological indices via M-polynomial of DM.Figure 2
Chemical graph of dexamethasoneDM.Theorem 10.
LetDM be dexamethasone, then M-polynomial of DM is MDMu,v=uv2+4uv3+4uv4+2u2v2+9u2v3+2u2v4+u3v3+6u3v4+2u4v4.Proof.
LetDM represent the dexamethasone, then by using Figures 1 and 2 and Table 1, we have that there are four partitions of the vertex set with respect to the vertex degree defined as Vn=x∈VDM,dx=n where n=1,2,3,4.
Figures1 and 2 and Table 2 show that there are nine partitions of the edge set with respect to the degree of end-vertices of the edge. These partitions are represented with En, where En=e=xy∈EDM,dx=a,dy=b where n=1,2,3,…,9 and a,b=1,2,3,4 with a≤b. We have the following result by using the definition of M-polynomial:(20)MDMu,v=∑ψ≤j≤k≤ΨmjkDMujvk=∑1≤j≤k≤4mjkDMujvk,MDMu,v=∑1≤2m12DMu1v2+∑1≤3m13DMu1v3+∑1≤4m14DMu1v4+∑2≤2m22DMu2v2+∑2≤3m23DMu2v3+∑2≤4m24DMu2v4+∑3≤3m33DMu3v3+∑3≤4m34DMu3v4+∑3≤4m44DMu4v4,MDMu,v=E1DMu1v2+E2DMu1v3+E3DMu1v4+E4DMu2v2+E5DMu2v3+E6DMu2v4+E2DMu3v3+E7DMu3v4+E8DMu4v4,MDMu,v=uv2+4uv3+4uv4+2u2v2+9u2v3+2u2v4+u3v3+6v3v4+2u4v4.
The plot of the M-polynomial ofDM is shown in Figure 3.Table 1
Vertex partition ofDM.
dx1234Number of vertices9874Table 2
Edge partition ofDM.
dx,dy1,21,31,42,22,32,43,33,44,4Number of edges144292162Figure 3
3D plot of M-polynomial of dexamethasoneDM.


## 4. Topological Indices of Dexamethasone

Theorem 11.
LetDM represent the dexamethasone and(21)MDMu,v=uv2+4uv3+4uv4+2u2v2+9u2v3+2u2v4+u3v3+6u3v4+2u4v4.Then,(1)
RRRDM=10+92+23+66(2)
AG1DM=1/12120+272+743+456(3)
SKDM=84(4)
SK1DM=221/2(5)
SK2DM=240(6)
EM1DM=412(7)
SCIDM=1/210630+1052+703+5465+1056+1807(8)
SCIλDM=3λ+6⋅4λ+13⋅5λ+3⋅6λ+6⋅7λ+2⋅8λ(9)
ReZG3DM=1346Proof.
(22)Qu−1MDMu,v=v2+4v3+4v4+2uv2+9uv3+2uv4+u2v3+6u2v4+2u3v4,Qv−1Qu−1MDMu,v=v+4v2+4v3+2uv+9uv2+2uv3+u2v2+6u2v3+2u3v3,Dv1/2Qv−1Qu−1MDM=v+42v2+43v3+2uv+92uv2+23uv3+2u2v2+63u2v3+23u3v3,Du1/2Dv1/2Qv−1Qu−1MDM=2uv+92uv2+23uv3+2u2v2+66u2v3+6u3v3,Sv1/2MDMu,v=22uv2+433uv3+2uv4+2u2v2+33u2v3+u2v4+33u3v3+3u3v4+u4v4,Su1/2Sv1/2MDMu,v=22uv2+433uv3+2uv4+u2v2+362u2v3+22u2v4+13u3v3+3u3v4+12u4v4,JSu1/2Sv1/2MDMu,v=22u3+3+433u4+4+362u5+2+326u6+3u7+12u8,DuJSu1/2Sv1/2MDMu,v=322u3+43+433u4+54+362u5+2+32u6+73u7+4u8,12DuJSu1/2Sv1/2MDMu,v=324u3+23+433u4+54+364u5+2+322u6+732u7+2u8,DuMDMu,v=uv2+4uv3+4uv4+4u2v2+18u2v3+4u2v4+3u3v3+18u3v4+8u4v4,DvMDMu,v=2uv2+12uv3+16uv4+4u2v2+27u2v3+8u2v4+3u3v3+24u3v4+8u4v4,Du+DvMDMu,v=3uv2+16uv3+20uv4+8u2v2+45u2v3+12u2v4+6u3v3+42u3v4+16u4v4,12Du+DvMDMu,v=32uv2+8uv3+10uv4+4u2v2+452u2v3+6u2v4+3u3v3+21u3v4+8u4v4,DuDvMDMu,v=2uv2+12uv3+16uv4+8u2v2+54u2v3+16u2v4+9u3v3+72u3v4+32u4v4,12DuDvMDMu,v=uv2+6uv3+8uv4+4u2v2+27u2v3+8u2v4+92u3v3+36u3v4+16u4v4,JMDMu,v=u3+6u4+13u5+3u6+6u7+2u8,Du2JMDMu,v=9u3+96u4+325u5+108u6+294u7+128u8,14Du2JMDMu,v=94u3+24u4+3254u5+27u6+1472u7+32u8,Qu−2JMDMu,v=u+6u2+13u3+3u4+6u5+2u6,Du2Qu−2JMDMu,v=u+24u2+117u3+48u4+150u5+72u6,Su1/2JMDMu,v=33u3+3u4+1355u5+62u6+677u7+22u8,DuλJMDMu,v=3λu3+6⋅4λu4+13⋅5λu5+3⋅6λu6+6⋅7λu7+2⋅8λu8,DvDu+DvMDMu,v=6uv2+48uv3+80uv4+16u2v2+135u2v3+48u2v4+18u3v3+168u3v4+64u4v4,DuDvDu+DvMDMu,v=6uv2+48uv3+80uv4+32u2v2+270u2v3+96u2v4+54u3v3+504u3v4+256u4v4.(1)
RRRDM=Du1/2Dv1/2Qu−1Qv−1MDMu,v|u=v=1=10+92+23+66(2)
AG1DM=1/2DuJSu1/2Su1/2MDMu,v|u=1=1/12120+272+743+456(3)
SKDM=1/2Du+DvMDMu,v|u=v=1=84(4)
SK1DM=1/2DuDvMDMu,v|u=v=1=221/2(5)
Sk2DM=1/4Du2JMDMu,v|u=1=240(6)
EM1DM=Du2Qu−2JMDMu,v|u=1=412(7)
SCIDM=Su1/2JMDMu,v|u=1=1/210630+1052+703+5465+1056+1807(8)
SCIλDM=DuλJMDMu,v|u=1=3λ+6⋅4λ+13⋅5λ+3⋅6λ+6⋅7λ+2⋅8λ(9)
ReZG3DM=DuDvDu+DvMDMu,v|u=v=1=1346
The results of topological indices are plotted in Figure4.Figure 4
Plot of topological indices of dexamethasoneDM.


## 5. Conclusion

We have presented the proofs of the formulas of some topological indices, which are derived from M-polynomial. The structure of the molecule can explore the chemical and biological behavior of the chemical compound. In the present work, we gave the computational analysis of the dexamethasone used during the treatment of COVID-19 by finding the topological indices. These topological indices are calculated from the M-polynomial of dexamethasone.



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