Invariants for Chemical Structures Used in the Treatment of COVID-19

The world is going through the pandemic of COVID-19. Remdesivir, chloroquine, hydroxychloroquine and theaflavin are some drugs currently in use for the treatment of the COVID-19 patients. This manuscript aims to compute the irregularity measures for these chemical structures. We believe that our results can be utilized to improve the capacity of these chemical structures.


Introduction
Historically, the epidemics of influenza and cholera were regarded as the most devastating. Recently, the COVID-19 pandemic which originated from China has affected the health of people all around the world [1]. By September 2020, there were over 32.7 million confirmed cases of COVID-19 including 991,000 deaths. According to WHO, no particular medication for this disease is available. So, it's very important to choose suitable medicine for its treatment. Researchers have tested some chemical compounds [2, 3,4,5,6] and obtained favorable results. These compounds include chloroquine, hydroxylchloroquine, remdesivir (GS5734), and theaflavin.
Molecular graph is a [7,8] simple graph in which the vertices and edges are considered as the atoms and bonds of the compound, respectively.
Let G = (V,E) be a simple connected graph, where V is the set of vertices and E represents the number of edges present in the graph. The degree of vertex depicts the number of edges attached with that vertex and it is denoted by dv, where v ∈ V (G) and e represents an edge e = uv ∈ E(G). Topological indices (TIs) help us to describe the structure of the graph [9,10,11,12,13,14,15,16,17,18]. The first ever TI was Scientific Inquiry and Review Volume 4 Issue 3, 2020 presented by Wiener in 1947 [19], when he was trying to find out the boiling point of alkanes. Gutman, in 1975, presented the definition [20] of Zagreb indices. the oldest degree-based descriptors are two indices known as the first and second Zagreb indices, respectively. The properties of these indices were investigated extensively [21,22]. These properties can be represented by the mathematical formulae given below.
TI is recognized as an irregularity index [23] if certain conditions are met. Firstly, if the TI of a graph is greater than zero or equal to zero. Secondly, if the TI of a graph is equal to zero, given the graph is regular. Table 1 shows the irregularity indices.

IRF G MG
School of Science Volume 4 Issue 3, 2020

Irregularities of the Chemical Structures used for the Treatment of the Patients of COVID-19
In this section, we discuss some chemical structures used for the treatment and management of the patients of COVID-19 by means of the chemical graph theory. We associate these structures with a molecular graph. We can observe from these two-dimensional molecular structures that all of them belong to the family of the irregular graph, so the best way to express these irregularities are irregularity indices. Here, we discuss sixteen irregularity indices for Scientific Inquiry and Review Volume 4 Issue 3, 2020 chloroquine, hydroxychloroquine, remdesivir, and theaflavin in four subsections.

Irregularity Indices for Remdesivir
This section covers the irregularity indices of remdesivir. Figure 1 shows the two dimensional molecular graph of remdesivir. The degree-based edge partition of remdesivir is given in Table 2 Table 2. Partition of E (Remdesivir) Theorem 2.1. Let's suppose G is the graph of remdesivir. The irregularity indices are as follows: (1) V AR(G) = 13.60.

Irregularity Indices for Chloroquine
This section focuses on the irregularity indices for chloroquine. The two-dimensional molecular graph of chloroquine (Figure 2) has three types of vertices.  Table 3 shows the degree-based edge partition of chloroquine (1) V AR(G) = 0.44.
(2) AL(G) = 18.    Proof. Using the edge partition of chloroquine given in Table 3, the irregularities of chloroquine are computed as follows: Scientific Inquiry and Review Volume 4 Issue 3, 2020 = 8.37.

Irregularities of Hydroxychloroquine
In this section, irregularity indices for hydroxychloroquine are discussed. The two-dimensional molecular graph of hydroxychloroquine is given in Figure 3. There are three types of vertices present in the graph. The degree-based edge partition of hydroxychloroquine is given below in Table 4. (1) V AR(G) = 0.43.
(2) AL(G) = 18.              Proof. Using the edge partition of hydroxychloroquine given in Table  4, the irregularities of hydroxychloroquine are computed as follows:

Irregularities of Theaflavin
In this section, irregularity indices for theaflavin are computed. The two-dimensional molecular graph of theaflavin ( Figure 4) depicts that three types of vertices are present in the graph.  Table 5 shows the degree-based edge partition of theaflavin. (1) V AR(G) = 0.67.

Conclusion
TIs are very helpful in predicting the physicochemical properties of chemical structures without involving laboratory work. This paper discusses the irregularity indices for chemical structures used in the treatment of COVID-19 namely chloroquine, hydroxychloroquine, remdesivir, and theaflavin. All these chemical structures have an irregular structure, so the best way of predicting their properties is through irregularity indices.