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A graphical index is a numeric value corresponding to a graph which is structurally invariant and in molecular graph theory these invariants are known as topological indices. In the field of Chemical and Medical Sciences, the topological indices are used to study the chemical, biological, medical and pharmaceutical features of drugs. With reference to the previous deadly diseases, the COVID-19 pandemic has considered to be the biggest life threatening issue that modern medicines have ever tackled. COVID-19 is immedicable and even the existing treatments are only helping the certain group of sufferers. Scientists have tested available antiviral agents and got a favorable impact on recovering from pandemic. Some of these antiviral agents are remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. Keeping in view of the importance of topological indices in the study of pharmaceutical and chemical drugs, in this paper, we calculate the
*M _{dn}*-Polynomial, some downhill Zagreb topological indices and some downhill Zagreb polynomials of some of the anti viral agents remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. The results thus obtained may be useful for the finding new medicine and vaccine for the treatment of COVID-19.

In the period of dynamic technical development, the large number of drugs and pharmaceutical techniques has been emerging every year which requires the massive amount of work to examine the biological and chemical properties of these drugs. Also, lots of experiments have to be done on these new drugs to find out the side effects and benefits on human body. These heavy works loaded experiments in laboratories may affect the impoverished countries especially Africa and Southeast Asia. During the initial stages of chemical experiments, the scientists have compared the structure of the compounds and its experimental values and pointed that they are closely related [

In the modeling of medical mathematics, the structure of medicine is considered as an undirected graph, where the vertices and edges are considered to be atoms and the chemical bonds respectively. The information pertaining physiochemical properties and the biological activities of molecular graph of compounds are important in pharmaceutical drug design. These properties can be anticipated without any use of laboratories but by a conventional aid of chemical graph theory known as the topological index. A graphical index is a numeric value corresponding to a graph which is structurally invariant and in molecular graph theory these invariants are known as topological indices. The first and second Zagreb indices are extensively studied among the various classes of topological indices and have many applications in the molecular graph theory. The Zagreb indices play a vital role in the theory of total π-electron energy of alternant hydrocarbons. Gutman and Trinajstic introduced the first and second Zagreb indices in 1972 [

With reference to the previous deadly diseases, the COVID-19 pandemic has considered to be the biggest life threatening issue that modern medicines have ever tackled. The scientists and doctors have been working tirelessly in finding the drugs which may save the sufferers and may even protect them from getting affected. As of 26th August 2020, there were more than 24 million reported resulting in 819,000 deaths and 16,620,943 have been recovered across 188 countries and territories (from world meters information). COVID-19 is immedicable and even the existing treatments are only helping the certain group of sufferers. No treatment has been fully licensed by the food and the drug administration agency for COVID-19.

Scientists have tested some of the available antiviral agents and got a favorable impact on recovering from pandemic by using remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. The first drug to get the emergency approval from food and drug administration for the use of COVID-19 is remdesivir. It ceases the reproduction of the virus. This drug was initially used as an antiviral agent for Hepatitis C and Ebola. From the preliminary trials, it has been observed that the drug can reduce the recovery time of the COVID-19 sufferers from 15 days to 11 days. In 1930s the German scientists incorporated chloroquine as a drug against malaria. In 1946, the scientists invented the less toxic version of chloroquine called hydroxychloroquine and later the drug was approved for other diseases also. During the initial stages of the deadly pandemic, the scientists have found that both chloroquine and hydroxychloroquine can control the virus from reproducing the cells [

Let G = ( V , E ) be a graph of order n = | V | . The open neighborhood of a vertex v ∈ V is the set N ( v ) = { u | u v ∈ E } , while the closed neighborhood is the set N [ v ] = N ( v ) ∪ { v } . Each vertex in u ∈ N ( v ) is called a neighbor of v, and | N ( v ) | is called the degree of v, and denoted d e g ( v ) . Any terminology in graph theory not defined here, we refer the reader to [

Definition 2.1. [

Definition 2.2. A vertex v is downhill dominates a vertex u if there exists a downhill path originated from v to u. The downhill neighborhood of a vertex v is denotes by N d n ( v ) and define as

N d n ( v ) = { u : v down hill dominates u } .

The downhill degree of the vertexv, denotes by d d n ( v ) , is the number of downhill neighbors of v, that means d d n ( v ) = | N d n ( v ) | .

Definition 2.3. [

D W M 1 ( G ) = ∑ v ∈ V ( G ) ( d d n ( v ) ) 2 ,

D W M 2 ( G ) = ∑ v u ∈ E ( G ) d d n ( v ) d d n ( u )

and

D W F ( G ) = ∑ v ∈ V ( G ) ( d d n ( v ) ) 3 .

Definition 2.4. [

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 ,

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u )

and

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 .

Definition 2.5. The M_{dn}-polynomial of G is defined as

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j ,

where δ d n = Min { d d n ( v ) | v ∈ V ( G ) } , Δ d n = Max { d d n ( v ) | v ∈ V ( G ) } and m i , j is the number of edges v u ∈ E ( G ) such that { d d n ( v ) , d d n ( u ) } = { i , j } and i ≤ j .

Definition 2.6. Let G = ( V , E ) be a graph. Then the first, second and forgotten downhill modified Zagreb indices are defined by

D W M 1 * ( G ) = ∑ v u ∈ E ( G ) d d n ( v ) + d d n ( u )

and

D W F * ( G ) = ∑ v u ∈ E ( G ) ( d d n ( v ) ) 2 + ( d d n ( u ) ) 2 .

_{dn}-polynomial with the following reserved notations

D x = x ∂ ( f ( x ) ) ∂ x , D y = y ∂ ( f ( y ) ) ∂ y .

We associated the graphs with the chemical structures of remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone where atoms are represented by vertices and chemical bonds are represented by edges. Then by using the symmetry of the molecular structures of remdesivir, chloroquine, hydroxychloroquine,

Topological Index | Derivation from DZP | Topological Index | Derivation from M d n ( G , x , y ) |
---|---|---|---|

D W M 1 ( G ) | D x ( D W M 1 ( G , x ) ) | x = 1 | D W M 1 * ( G ) | ( D x + D y ) ( M d n ( G , x , y ) ) | x = y = 1 |

D W M 2 ( G ) | D x ( D W M 2 ( G , x ) ) | x = 1 | D W F * ( G ) | ( D x 2 + D y 2 ) ( M d n ( G , x , y ) ) | x = y = 1 |

D W F ( G ) | D x ( D W F ( G , x ) ) | x = 1 |

theaflavin and dexamethasone we counted the edges and vertices by a simple counting method. By applying the formula of the polynomial, we derived the downhill Zagreb polynomials and M_{dn}-polynomial of remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. From these downhill Zagreb polynomials and M_{dn}-polynomial we recovered 5 downhill degree-based topological indices by using Derivation. We used Matlab 2017 to plot our results.

In this section, we give our main computational results. We compute downhill Zagreb polynomials and M_{dn}-polynomial of molecular graph of remdesivir.

Theorem 4.1. Let G be the molecular graph of remdesivir. Then,

D W M 1 ( G , x ) = x 361 + 4 x 81 + 3 x 49 + 4 x 36 + x 25 + 5 x 16 + 3 x 4 + 8 x + 12 ,

D W M 2 ( G , x ) = x 171 + x 133 + 3 x 81 + 2 x 49 + x 36 + 2 x 24 + 2 x 18 + 4 x 16 + 2 x 9 + x 7 + 4 x 6 + x 5 + 2 x 4 + 3 x + 15 ,

D W F ( G , x ) = x 6859 + 4 x 729 + 3 x 343 + 4 x 216 + x 125 + 5 x 64 + 3 x 8 + 8 x + 12.

Proof. Let G be the molecular graph of remdesivir (

Then, D W M 1 ( G , x ) is obtained as follows.

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 = x 19 2 + 4 x 9 2 + 3 x 7 2 + 4 x 6 2 + x 5 2 + 5 x 4 2 + 3 x 2 2 + 8 x 1 2 + 12 x 0 2 = x 361 + 4 x 81 + 3 x 49 + 4 x 36 + x 25 + 5 x 16 + 3 x 4 + 8 x + 12.

The graph G has 44 edges. Suppose that E a , b = { u v ∈ E ( G ) : d d n ( u ) = a and d d n ( v ) = b } and | E a , b | = m i , j . In a graph G there are 20 types of edges based on the downhill degree of the vertices of each edge. From

Then, D W M 2 ( G , x ) is obtained as follows.

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u ) = | E 9 , 19 | x 9 × 19 + | E 7 , 19 | x 7 × 19 + | E 9 , 9 | x 9 × 9 + | E 2 , 9 | x 2 × 9 + | E 1 , 9 | x 1 × 9 + | E 7 , 7 | x 7 × 7 + | E 1 , 7 | x 1 × 7 + | E 6 , 6 | x 6 × 6 + | E 4 , 6 | x 4 × 6 + | E 1 , 6 | x 1 × 6 + | E 1 , 5 | x 1 × 5 + | E 4 , 4 | x 4 × 4 + | E 2 , 2 | x 2 × 2 + | E 1 , 1 | x 1 × 1 + | E 0 , 19 | x 0 × 19 + | E 0 , 9 | x 0 × 9 + | E 0 , 7 | x 0 × 7 + | E 0 , 6 | x 0 × 6 + | E 0 , 5 | x 0 × 5 + | E 0 , 1 | x 0 × 1

= x 171 + x 133 + 3 x 81 + 2 x 49 + x 36 + 2 x 24 + 2 x 18 + 4 x 16 + 2 x 9 + x 7 + 4 x 6 + x 5 + 2 x 4 + 3 x + 15.

Now, we calculate D W F ( G , x ) similar to D W M 1 ( G , x ) , then

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 = x 19 3 + 4 x 9 3 + 3 x 7 3 + 4 x 6 3 + x 5 3 + 5 x 4 3 + 3 x 2 3 + 8 x 1 3 + 12 x 0 3 = x 6859 + 4 x 729 + 3 x 343 + 4 x 216 + x 125 + 5 x 64 + 3 x 8 + 8 x + 12.

Theorem 4.2. Let G be the molecular graph of remdesivir. Then,

M d n ( G , x , y ) = x 9 y 19 + x 7 y 19 + 3 x 9 y 9 + 2 x 2 y 9 + 2 x y 9 + 2 x 7 y 7 + x y 7 + x 6 y 6 + 2 x 4 y 6 + 4 x y 6 + x y 5 + 4 x 4 y 4 + 2 x 2 y 2 + 3 x y + 2 y 19 + y 9 + 3 y 7 + 4 y 6 + 3 y 5 + 2 y .

Proof. From Theorem 4.1 and using the definition M d n ( G , x , y ) , we have

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j = | E 9 , 19 | x 9 y 19 + | E 7 , 19 | x 7 y 19 + | E 9 , 9 | x 9 y 9 + | E 2 , 9 | x 2 y 9 + | E 1 , 9 | x y 9 + | E 7 , 7 | x 7 y 7 + | E 1 , 7 | x y 7 + | E 6 , 6 | x 6 y 6 + | E 4 , 6 | x 4 y 6 + | E 1 , 6 | x y 6 + | E 1 , 5 | x y 5 + | E 4 , 4 | x 4 y 4 + | E 2 , 2 | x 2 y 2 + | E 1 , 1 | x y + | E 0 , 19 | x 0 y 19 + | E 0 , 9 | x 0 y 9 + | E 0 , 7 | x 0 y 7 + | E 0 , 6 | x 0 y 6 + | E 0 , 5 | x 0 y 5 + | E 0 , 1 | x 0 y

= x 9 y 19 + x 7 y 19 + 3 x 9 y 9 + 2 x 2 y 9 + 2 x y 9 + 2 x 7 y 7 + x y 7 + x 6 y 6 + 2 x 4 y 6 + 4 x y 6 + x y 5 + 4 x 4 y 4 + 2 x 2 y 2 + 3 x y + 2 y 19 + y 9 + 3 y 7 + 4 y 6 + 3 y 5 + 2 y .

_{dn}-polynomial of remdesivir.

Now using the Theorem 4.1, we calculate the first, second and forgotten downhill Zagreb indices of the molecular graph of remdesivir in the following theorem.

Theorem 4.3. Let G be the molecular graph of remdesivir. Then,

D W M 1 ( G ) = 1101 ,

D W M 2 ( G ) = 894 ,

D W F ( G ) = 12145.

Proof. Let

D W M 1 ( G , x ) = f ( x ) = x 361 + 4 x 81 + 3 x 49 + 4 x 36 + x 25 + 5 x 16 + 3 x 4 + 8 x + 12 .

Then

D x f ( x ) = 361 x 361 + 324 x 81 + 147 x 49 + 144 x 36 + 25 x 25 + 80 x 16 + 12 x 4 + 8 x .

From

D W M 1 ( G ) = D x f ( x ) | x = 1 = 1101.

For D W M 2 ( G ) , let

D W M 2 ( G , x ) = h ( x ) = x 171 + x 133 + 3 x 81 + 2 x 49 + x 36 + 2 x 24 + 2 x 18 + 4 x 16 + 2 x 9 + x 7 + 4 x 6 + x 5 + 2 x 4 + 3 x + 15.

Then

D x h ( x ) = 171 x 171 + 133 x 133 + 243 x 81 + + 98 x 49 + 36 x 36 + 48 x 24 + 36 x 18 + 64 x 16 + 18 x 9 + 7 x 7 + 24 x 6 + 5 x 5 + + 8 x 4 + 3 x .

From

D W M 2 ( G ) = D x h ( x ) | x = 1 = 894.

Now, for D W F ( G ) . Let

D W F ( G , x ) = g ( x ) = x 6859 + 4 x 729 + 3 x 343 + 4 x 216 + x 125 + 5 x 64 + 3 x 8 + 8 x + 12.

Then

D x g ( x ) = 6859 x 6859 + 2916 x 729 + 1029 x 343 + 864 x 216 + 125 x 125 + 320 x 64 + 24 x 8 + 8 x .

From

D W F ( G ) = D x g ( x ) | x = 1 = 12145.

Now using the Theorem 4.2, we calculate the first, second and forgotten downhill modified Zagreb indices of the molecular graph of remdesivir in the following theorem.

Theorem 4.4. Let G be the molecular graph of remdesivir. Then,

D W M 1 * ( G ) = 407 ,

D W F * ( G ) = 3589.

Proof. Let

M d n ( G , x , y ) = f ( x , y ) = x 9 y 19 + x 7 y 19 + 3 x 9 y 9 + 2 x 2 y 9 + 2 x y 9 + 2 x 7 y 7 + x y 7 + x 6 y 6 + 2 x 4 y 6 + 4 x y 6 + x y 5 + 4 x 4 y 4 + 2 x 2 y 2 + 3 x y + 2 y 19 + y 9 + 3 y 7 + 4 y 6 + 3 y 5 + 2 y .

Then

( D x + D x ) f ( x , y ) = 28 x 9 y 19 + 26 x 7 y 19 + 54 x 9 y 9 + 22 x 2 y 9 + 20 x y 9 + 28 x 7 y 7 + 8 x y 7 + 12 x 6 y 6 + 20 x 4 y 6 + 28 x y 6 + 6 x y 5 + 32 x 4 y 4 + 8 x 2 y 2 + 6 x y + 38 y 19 + 9 y 9 + 21 y 7 + 24 y 6 + 15 y 5 + 2 y ,

( D x 2 + D y 2 ) f ( x , y ) = 442 x 9 y 19 + 410 x 7 y 19 + 486 x 9 y 9 + 170 x 2 y 9 + 164 x y 9 + 196 x 7 y 7 + 50 x y 7 + 72 x 6 y 6 + 140 x 4 y 6 + 148 x y 6 + 26 x y 5 + 128 x 4 y 4 + 16 x 2 y 2 + 6 x y + 722 y 19 + 81 y 9 + 147 y 7 + 144 y 6 + 75 y 5 + 2 y .

Using

D W M 1 * ( G ) = ( D x + D x ) f ( x , y ) | x = y = 1 = 407 ,

D W F * ( G ) = ( D x 2 + D y 2 ) f ( x , y ) | x = y = 1 = 3589.

We evaluate the downhill Zagreb polynomials and M_{dn}-polynomial of the molecular graph of chloroquine in the following two theorems.

Theorem 4.5. Let G be the molecular graph of chloroquine. Then,

D W M 1 ( G , x ) = 3 x 81 + x 49 + x 25 + x 16 + 6 x 4 + 4 x + 6 ,

D W M 2 ( G , x ) = 2 x 81 + 2 x 18 + x 14 + x 10 + x 9 + 2 x 7 + 5 x 4 + x + 8 ,

D W F ( G , x ) = 3 x 729 + x 343 + x 125 + x 64 + 6 x 8 + 4 x + 6.

Proof. Let G be the molecular graph of chloroquine (

one vertex of downhill degree 5, one vertex of downhill degree 4, 6 vertices of downhill degree 2, 4 vertices of downhill degree 1 and 6 vertices of downhill degree 0.

Then, D W M 1 ( G , x ) is obtained as follows.

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 = 3 x 9 2 + x 7 2 + x 5 2 + x 4 2 + 6 x 2 2 + 4 x 1 2 + 6 x 0 2 = 3 x 81 + x 49 + x 25 + x 16 + 6 x 4 + 4 x + 6.

The graph G has 23 edges. In a graph G there are 13 types of edges based on the downhill degree of the vertices of each edge. From

Then, D W M 2 ( G , x ) is obtained as follows.

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u ) = | E 9 , 9 | x 9 × 9 + | E 2 , 9 | x 2 × 9 + | E 1 , 9 | x 1 × 9 + | E 2 , 7 | x 2 × 7 + | E 1 , 7 | x 1 × 7 + | E 2 , 5 | x 2 × 5 + | E 1 , 4 | x 1 × 4 + | E 2 , 2 | x 2 × 2 + | E 1 , 1 | x 1 × 1 + | E 0 , 9 | x 0 × 9 + | E 0 , 5 | x 0 × 5 + | E 0 , 4 | x 0 × 4 + | E 0 , 1 | x 0 × 1 = 2 x 81 + 2 x 18 + x 14 + x 10 + x 9 + 2 x 7 + 5 x 4 + x + 8.

Now, we calculate D W F ( G , x ) similar to D W M 1 ( G , x ) , then

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 = 3 x 9 3 + x 7 3 + x 5 3 + x 4 3 + 6 x 2 3 + 4 x 1 3 + 6 x 0 3 = 3 x 729 + x 343 + x 125 + x 64 + 6 x 8 + 4 x + 6.

Theorem 4.6. Let G be the molecular graph of chloroquine. Then,

M d n ( G , x , y ) = 2 x 9 y 9 + 2 x 2 y 9 + x y 9 + x 2 y 7 + 2 x y 7 + x 2 y 5 + x y 4 + 4 x 2 y 2 + x y + 2 y 9 + 2 y 5 + 2 y 4 + 2 y .

Proof. From Theorem 4.5 and using the definition M d n ( G , x , y ) , we have

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j = | E 9 , 9 | x 9 y 9 + | E 2 , 9 | x 2 y 9 + | E 1 , 9 | x y 9 + | E 2 , 7 | x 2 y 7 + | E 1 , 7 | x y 7 + | E 2 , 5 | x 2 y 5 + | E 1 , 4 | x 1 y 4 + | E 2 , 2 | x 2 y 2 + | E 1 , 1 | x y + | E 0 , 9 | x 0 y 9 + | E 0 , 5 | x 0 y 5 + | E 0 , 4 | x 0 y 4 + | E 0 , 1 | x 0 y = 2 x 9 y 9 + 2 x 2 y 9 + x y 9 + x 2 y 7 + 2 x y 7 + x 2 y 5 + x y 4 + 4 x 2 y 2 + x y + 2 y 9 + 2 y 5 + 2 y 4 + 2 y .

_{dn}-polynomial of chloroquine.

Now using the Theorem 4.5, we calculate the first, second and forgotten downhill Zagreb indices of the molecular graph of chloroquine in the following theorem.

Theorem 4.7. Let G be the molecular graph of chloroquine. Then,

D W M 1 ( G ) = 361 ,

D W M 2 ( G ) = 266 ,

D W F ( G ) = 2771.

Proof. The proof similarly to the proof of Theorem 4.3.

Now using the Theorem 4.6, we calculate the first, second and forgotten downhill modified Zagreb indices of the molecular graph of chloroquine in the following theorem.

Theorem 4.8. Let G be the molecular graph of chloroquine. Then,

D W M 1 * ( G ) = 161 ,

D W F * ( G ) = 1055.

Proof. The proof similarly to the proof of Theorem 4.4.

We evaluate the downhill Zagreb polynomials and M_{dn}-polynomial of the molecular graph of hydroxychloroquine in the following two theorems.

Theorem 4.9. Let G be the molecular graph of hydroxychloroquine. Then,

D W M 1 ( G , x ) = 3 x 81 + x 64 + x 25 + x 16 + 8 x 4 + 3 x + 6 ,

D W M 2 ( G , x ) = 2 x 81 + 2 x 18 + 2 x 16 + x 10 + x 9 + x 8 + 6 x 4 + x + 8 ,

D W F ( G , x ) = 3 x 729 + x 512 + x 125 + x 64 + 8 x 8 + 3 x + 6.

Proof. Let G be the molecular graph of hydroxychloroquine (

Then, D W M 1 ( G , x ) is obtained as follows.

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 = 3 x 9 2 + x 8 2 + x 5 2 + x 4 2 + 8 x 2 2 + 3 x 1 2 + 6 x 0 2 = 3 x 81 + x 64 + x 25 + x 16 + 8 x 4 + 3 x + 6.

The graph G has 24 edges. In a graph G there are 14 types of edges based on the downhill degree of the vertices of each edge. From

Then, D W M 2 ( G , x ) is obtained as follows.

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u ) = | E 9 , 9 | x 9 × 9 + | E 2 , 9 | x 2 × 9 + | E 1 , 9 | x 1 × 9 + | E 2 , 8 | x 2 × 8 + | E 1 , 8 | x 1 × 8 + | E 2 , 5 | x 2 × 5 + | E 1 , 4 | x 1 × 4 + | E 2 , 2 | x 2 × 2 + | E 1 , 1 | x 1 × 1 + | E 0 , 9 | x 0 × 9 + | E 0 , 5 | x 0 × 5 + | E 0 , 4 | x 0 × 4 + | E 0 , 2 | x 0 × 2 + | E 0 , 1 | x 0 × 1 = 2 x 81 + 2 x 18 + 2 x 16 + x 10 + x 9 + x 8 + 6 x 4 + x + 8.

Now, we calculate D W F ( G , x ) similar to D W M 1 ( G , x ) , then

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 = 3 x 9 3 + x 8 3 + x 5 3 + x 4 3 + 8 x 2 3 + 3 x 1 3 + 6 x 0 3 = 3 x 729 + x 512 + x 125 + x 64 + 8 x 8 + 3 x + 6.

Theorem 4.10. Let G be the molecular graph of hydroxychloroquine. Then,

M d n ( G , x , y ) = 2 x 9 y 9 + 2 x 2 y 9 + x y 9 + 2 x 2 y 8 + x y 8 + x 2 y 5 + x y 4 + 5 x 2 y 2 + x y + 2 y 9 + 2 y 5 + 2 y 4 + y 2 + y .

Proof. From Theorem 4.9 and using the definition M d n ( G , x , y ) , we have

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j = | E 9 , 9 | x 9 y 9 + | E 2 , 9 | x 2 y 9 + | E 1 , 9 | x y 9 + | E 2 , 8 | x 2 y 8 + | E 1 , 8 | x y 8 + | E 2 , 5 | x 2 y 5 + | E 1 , 4 | x 1 y 4 + | E 2 , 2 | x 2 y 2 + | E 1 , 1 | x y + | E 0 , 9 | x 0 y 9 + | E 0 , 5 | x 0 y 5 + | E 0 , 4 | x 0 y 4 + | E 0 , 2 | x 0 y 2 + | E 0 , 1 | x 0 y = 2 x 9 y 9 + 2 x 2 y 9 + x y 9 + 2 x 2 y 8 + x y 8 + x 2 y 5 + x y 4 + 5 x 2 y 2 + x y + 2 y 9 + 2 y 5 + 2 y 4 + y 2 + y .

_{dn}-polynomial of hydroxychloroquine.

Now by using the Theorem 4.9, we calculate the first, second and forgotten downhill Zagreb indices of the molecular graph of hydroxychloroquine in the following theorem.

Theorem 4.11. Let G be the molecular graph of hydroxychloroquine. Then,

D W M 1 ( G ) = 383 ,

D W M 2 ( G ) = 282 ,

D W F ( G ) = 2955.

Proof. The proof similarly to the proof of Theorem 4.3.

Now, by using the Theorem 4.10, we can calculate the first, second and forgotten downhill modified Zagreb indices of the molecular graph of hydroxychloroquine in the following theorem.

Theorem 4.12. Let G be the molecular graph of hydroxychloroquine. Then,

D W M 1 * ( G ) = 170 ,

D W F * ( G ) = 1114.

Proof. The proof is similar to the proof of Theorem 4.4.

We evaluate the downhill Zagreb polynomials and M_{dn}-polynomial of the molecular graph of theaflavin in the following two results.

Theorem 4.13. Let G be the molecular graph of theaflavin. Then,

D W M 1 ( G , x ) = 9 x 324 + 9 x 49 + 2 x 9 + 12 ,

D W M 2 ( G , x ) = 8 x 324 + 6 x 49 + 32 ,

D W F ( G , x ) = 9 x 5832 + 9 x 343 + 2 x 27 + 12.

Proof. Let G be the molecular graph of theaflavin (

Then, D W M 1 ( G , x ) is obtained as follows.

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 = 9 x 18 2 + 9 x 7 2 + 2 x 3 2 + 12 x 0 2 = 9 x 324 + 9 x 49 + 2 x 9 + 12.

The graph G has 46 edges. In a graph G there are 5 types of edges based on the downhill degree of the vertices of each edge. From

Then, D W M 2 ( G , x ) is obtained as follows.

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u ) = | E 18 , 18 | x 18 × 18 + | E 7 , 7 | x 7 × 7 + | E 0 , 18 | x 0 × 18 + | E 0 , 7 | x 0 × 7 + | E 0 , 3 | x 0 × 3 = 8 x 324 + 6 x 49 + 32.

Now, we calculate D W F ( G , x ) similar to D W M 1 ( G , x ) , then

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 = 9 x 18 3 + 9 x 7 3 + 2 x 3 3 + 12 x 0 3 = 9 x 5832 + 9 x 343 + 2 x 27 + 12.

Theorem 4.14. Let G be the molecular graph of theaflavin. Then,

M d n ( G , x , y ) = 8 x 18 y 18 + 6 x 7 y 7 + 11 y 18 + 15 y 7 + 6 y 3 .

Proof. From Theorem 4.13 and using the definition M d n ( G , x , y ) , we have

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j = | E 18 , 18 | x 18 y 18 + | E 7 , 7 | x 7 y 7 + | E 0 , 18 | x 0 y 18 + | E 0 , 7 | x 0 y 7 + | E 0 , 3 | x 0 y 3 = 8 x 18 y 18 + 6 x 7 y 7 + 11 y 18 + 15 y 7 + 6 y 3 .

_{dn}-polynomial of theaflavin.

Now using the Theorem 4.13, we calculate the first, second and forgotten downhill Zagreb indices of the molecular graph of theaflavin in the following theorem.

Theorem 4.15. Let G be the molecular graph of theaflavin. Then,

D W M 1 ( G ) = 3375 ,

D W M 2 ( G ) = 2886 ,

D W F ( G ) = 55629.

Proof. The proof similarly to the proof of Theorem 4.3.

Now, by using the Theorem 4.14, we can calculate the first, second and forgotten downhill modified Zagreb indices of the molecular graph of theaflavin in the following result.

Theorem 4.16. Let G be the molecular graph of theaflavin. Then,

D W M 1 * ( G ) = 693 ,

D W F * ( G ) = 10125.

Proof. The proof similarly to the proof of Theorem 4.4.

We evaluate the downhill Zagreb polynomials and M_{dn}-polynomial of the molecular graph of dexamethasone in the following two results.

Theorem 4.17. Let G be the molecular graph of dexamethasone. Then,

D W M 1 ( G , x ) = 4 x 225 + 3 x 16 + 2 x 9 + 2 x 4 + 5 x + 12 ,

D W M 2 ( G , x ) = 2 x 225 + 2 x 60 + 2 x 45 + 2 x 30 + x 16 + x 15 + 2 x 4 + 2 x 3 + 2 x + 15 ,

D W F ( G , x ) = 4 x 3375 + 3 x 64 + 2 x 27 + 2 x 8 + 5 x + 12.

Proof. Let G be the molecular graph of dexamethasone (

Then, D W M 1 ( G , x ) is obtained as follows.

D W M 1 ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 2 = 4 x 15 2 + 3 x 4 2 + 2 x 3 2 + 2 x 2 2 + 5 x 1 2 + 12 x 0 2 = 4 x 225 + 3 x 16 + 2 x 9 + 2 x 4 + 5 x + 12.

The graph G has 31 edges. In a graph G there are 14 types of edges based on the downhill degree of the vertices of each edge. From

Then, D W M 2 ( G , x ) is obtained as follows.

D W M 2 ( G , x ) = ∑ v u ∈ E ( G ) x d d n ( v ) d d n ( u ) = | E 15 , 15 | x 15 × 15 + | E 4 , 15 | x 4 × 15 + | E 3 , 15 | x 3 × 15 + | E 2 , 15 | x 2 × 15 + | E 1 , 15 | x 1 × 15 + | E 4 , 4 | x 4 × 4 + | E 1 , 4 | x 1 × 4 + | E 1 , 3 | x 1 × 3 + | E 1 , 1 | x 1 × 1 + | E 0 , 15 | x 0 × 15 + | E 0 , 4 | x 0 × 4 + | E 0 , 3 | x 0 × 3 + | E 0 , 2 | x 0 × 2 + | E 0 , 1 | x 0 × 1 = 2 x 225 + 2 x 60 + 2 x 45 + 2 x 30 + x 16 + x 15 + 2 x 4 + 2 x 3 + 2 x + 15.

Now, we calculate D W F ( G , x ) similar to D W M 1 ( G , x ) , then

D W F ( G , x ) = ∑ v ∈ V ( G ) x ( d d n ( v ) ) 3 = 4 x 15 3 + 3 x 4 3 + 2 x 3 3 + 2 x 2 3 + 5 x 1 3 + 12 x 0 3 = 4 x 3375 + 3 x 64 + 2 x 27 + 2 x 8 + 5 x + 12.

Theorem 4.18. Let G be the molecular graph of dexamethasone. Then,

M d n ( G , x , y ) = 2 x 15 y 15 + 2 x 4 y 15 + 2 x 3 y 15 + 2 x 2 y 15 + x y 15 + x 4 y 4 + 2 x y 4 + 2 x y 3 + 2 x y + 5 y 15 + 3 y 4 + 2 y 3 + 4 y 2 + y .

Proof. From Theorem 4.17 and using the definition M d n ( G , x , y ) , we have

M d n ( G , x , y ) = ∑ δ d n ≤ i ≤ j ≤ Δ d n m i , j x i y j = | E 15 , 15 | x 15 y 15 + | E 4 , 15 | x 4 y 15 + | E 3 , 15 | x 3 y 15 + | E 2 , 15 | x 2 y 15 + | E 1 , 15 | x y 15 + | E 4 , 4 | x 4 y 4 + | E 1 , 4 | x 1 y 4 + | E 1 , 3 | x y 3 + | E 1 , 1 | x y + | E 0 , 15 | x 0 y 15 + | E 0 , 4 | x 0 y 4 + | E 0 , 3 | x 0 y 3 + | E 0 , 2 | x 0 y 2 + | E 0 , 1 | x 0 y = 2 x 15 y 15 + 2 x 4 y 15 + 2 x 3 y 15 + 2 x 2 y 15 + x y 15 + x 4 y 4 + 2 x y 4 + 2 x y 3 + 2 x y + 5 y 15 + 3 y 4 + 2 y 3 + 4 y 2 + y .

Now using the Theorem 4.17, we calculate the first, second and forgotten downhill Zagreb indices of the molecular graph of dexamethasone in the following theorem.

Theorem 4.19. Let G be the molecular graph of dexamethasone. Then,

D W M 1 ( G ) = 979 ,

D W M 2 ( G ) = 767 ,

D W F ( G ) = 13767.

Proof. The proof similarly to the proof of Theorem 4.3.

_{dn}-polynomial of dexamethasone.

Now, by using the Theorem 4.18, we can calculate the first, second and forgotten downhill modified Zagreb indices of the molecular graph of dexamethasone in the following theorem.

Theorem 4.20. Let G be the molecular graph of dexamethasone. Then,

D W M 1 * ( G ) = 316 ,

D W F * ( G ) = 3832.

Proof. The proof similarly to the proof of Theorem 4.4.

In this research work, some properties and calculations of the chemical compounds which are used for the treatment of COVID-19 in terms of first, second and forgotten downhill Zagreb indices and polynomials are obtained. In particular, remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. We evaluate some downhill Zagreb indices, M_{dn}-polynomial and some downhill Zagreb Polynomials of these structures with 3D graphical representation. As topological indices are very important to predict different properties and activities such as acentric factor, enthalpy, boiling point, critical pressure, entropy, etc. our results and calculations will be useful to maybe developing new drug and vaccine for the treatment of COVID-19.

The authors declare no conflicts of interest regarding the publication of this paper.

Al-Ahmadi, B., Saleh, A. and Al-Shammakh, W. (2021) Downhill Zagreb Topological Indices and M_{dn}-Polynomial of Some Chemical Structures Applied for the Treatment of COVID-19 Patients. Open Journal of Applied Sciences, 11, 395-413. https://doi.org/10.4236/ojapps.2021.114029