#### Abstract

Monkeypox has recently re-emerged. Since monkeypox has started to be seen in many countries without an epidemic, studies have started on new drugs used to treat this disease. Drug discovery is a challenging process. To alleviate this process, molecular descriptors used to predict the physical, chemical, and biological properties of chemical structures are studied. Molecular descriptors are numerical values of a chemical structure represented by a graph. In this study, M-polynomial and neighborhood M-polynomial molecular descriptors of drugs used and candidates for use in monkeypox are calculated and these descriptors are compared.

#### 1. Introduction

Monkeypox virus belongs to the *Orthopoxvirus* genus of the *Poxviridae* family. The disease was named monkeypox when it was first discovered in monkeys in 1958. In 1970, the first human case was seen a child in the Democratic Republic of the Congo. Monkeypox shows symptoms very similar to smallpox. Monkeypox is a viral zoonosis, that is, a virus transmitted from animals to humans. Symptoms of monkeypox include fever, swollen lymph nodes, blisters on the body, and crusty rashes on the body. People infected with this virus begin to show these symptoms between five and twenty-one days [1].

The World Health Organization (WHO) said that “Since 13 May 2022, cases of monkeypox have been reported to WHO from three WHO regions and 12 Member States that are not endemic for monkeypox virus” [1].

Although there is no specific vaccine or drug against monkeypox, historically, the smallpox vaccine has been shown to be protective against this disease. So, for the treatment of monkeypox, the MVA-BN vaccine in 2019 and the tecovirimat drug in 2022 were approved [1].

After the recent COVID-19 epidemic, the importance of rapid drug and vaccine discoveries has emerged. Drug discovery is not only a time-consuming process but also a costly process. In order to shorten this process, models that can predict the physicochemical and bioactivity properties of the newly produced molecular structure are obtained with topological indices studied in chemical mathematics [2–6]. These models are obtained via the structure-property/activity relationship (QSPR/QSAR) studies using molecular descriptors [7, 8].

Molecular descriptors are numerical descriptors of graphs obtained via chemical structures [9]. In 1947, Wiener estimated some properties of alkanes using the index [10]. Since then, many molecular descriptors have been studied to predict the properties of chemical structures without experimentation. Recently, M-polynomials and NM-polynomials have been defined and studied for many chemical structures (see [11–22]).

#### 2. Preliminaries

If graph has vertex set and edge set, then the order (size) of vertices (edges) is defined as (. The degree of the vertex of graph is denoted by . The neighborhood of a vertex in a graph *G*, , is the set of all vertices adjacent to [23]*.* Let and and .

*M*-polynomial of a graph is defined in [19] as

The neighborhood *M*-polynomial is defined in [20] as

Degree-based molecular descriptors and neighborhood degree-based molecular descriptors are, respectively, defined as

In this paper, the first Zagreb (), second Zagreb (), second modified Zagreb (), forgotten (), redefined third Zagreb (), symmetric division (), inverse sum indeg , harmonic (), and augmented Zagreb () indices are degree-based molecular descriptors, and third version of Zagreb (), neighborhood second modified Zagreb (), neighborhood second modified Zagreb (), neighborhood forgotten (), third NDe (), fifth NDe (), neighborhood inverse sum indeg (), neighborhood harmonic (), and Sanskruti () indices are neighborhood degree-based descriptors.

Table 1 shows molecular descriptors based on degrees and the sum of adjacent vertex degrees.

The following operators are used in Table 1:

If and , then . If and , then .

The used drugs for monkeypox are as follows.

Tecovirimat (also known as Tpoxx) is an antiviral medication. This drug is approved by the United States Food and Drug Administration (FDA) for the treatment of human smallpox disease. Moreover, it can be also used for the treatment of non-variola orthopoxviruses (including monkeypox) in an outbreak. Its chemical formula is C19H15F3N2O3. Figure 1 shows the structure of tecovirimat [24].

Cidofovir (also known as Vistide) is an antiviral medication. It is approved by the FDA for the treatment of cytomegalovirus retinitis in patients with acquired immunodeficiency syndrome (AIDS). Moreover, it is allowed for the use for the treatment of orthopoxviruses (including monkeypox) in an outbreak. Its chemical formula is C_{8}H_{14}N_{3}O_{6}P. Figure 2 shows the structure of cidofovir [24].

Brincidofovir (also known as Tembexa) (CMX001) is an antiviral medication that is a lipid conjugate of cidofovir. On June 4, 2021, it was approved by the FDA for the treatment of human smallpox disease in adult and paediatric patients, including neonates. This drug is currently developing for use in the treatment of monkeypox. Its chemical formula is C_{27}H_{52}N_{3}O_{7}P. Figure 3 shows the structure of brincidofovir [24].

In this study, the degree and neighborhood degree based on molecular descriptors of these molecules are computed, which will be useful in predicting the biological activity and physicochemical properties of new drugs that can be obtained with possible combinations of considered and used drugs for treatment of monkeypox.

#### 3. The *M*-Polynomials and NM-Polynomials of the Chemical Structures of Used and Potential Drugs against Monkeypox

The *M*-polynomials and NM-polynomials of the graphs obtained from tecovirimat, cidofovir, and brincidofovir drugs are found. Using these polynomials, various molecular descriptors depending on the degree and the sum of neighborhood vertex degrees for these structures are calculated.

Theorem 1. *If ** is the chemical graph of tecovirimat*, *then*

*Proof. *From Figure 1, the chemical graph of tecovirimat has 27 orders and 31 sizes, and also we can partition it as , , , , , . From equation (1),

Corollary 1. * Degree based on some molecular descriptors of the graph of tecovirimat:*(1)

*.*(2)

*.*(3)

*.*(4)

*.*(5)

*.*(6)

*.*(7)

*.*(8)

*.*(9)

*.*

*Proof. *From Table 1 and Theorem 1, we haveThe following results are obtained:(1).(2).(3).(4).(5).(6).(7).(8).(9).

Theorem 2. *The neighborhood M-polynomial of ** is*

*Proof. *From Figure 1, , , , , , , , , , , , , ,

From equation (2),

Corollary 2. * The molecular descriptors based on the neighborhood degree of graph are*(1)

*.*(2)

*.*(3)

*.*(4)

*.*(5)

*.*(6)

*.*(7)

*.*(8)

*.*(9)

*.*

*Proof. *Using Theorem 3, we obtain the following equations:The following results are obtained:(1).(2).(3).(4).(5).(6).(7)(8).(9).

Theorem 3. *If ** is the chemical graph of cidofovir*, *then*

*Proof. *From Figure 2, the chemical graph of cidofovir has 18 sizes and 18 orders. We can partition it as , , , , , , . From equation (1),

Corollary 3. * Degree based on some molecular descriptors of the chemical graph of cidofovir:*(1)

*.*(2)

*.*(3)

*.*(4)

*.*(5)

*.*(6)

*.*(7)

*.*(8)

*.*(9)

*.*

*Proof. *From Table 1 and Theorem 3, the following equations are obtained:

Theorem 4. *The neighborhood M-polynomial of ** is*

*Proof. *From Figure 2, the set of edges is divided as follows: , , , , , , , , ,

From (2),

Corollary 4. * The molecular descriptors based on the neighborhood degree of the graph are*(1)

*.*(2)

*.*(3)

*.*(4)

*.*(5)

*.*(6)

*.*(7)

*.*(8)

*.*(9)

*.*

*Proof. *Using Theorem 4, the following equations are obtained:

Theorem 5. *If ** is the chemical graph of brincidofovir*, *then*

*Proof. *From Figure 3, the chemical graph of brincidofovir has 38 sizes and 38 orders. We can partition it as , , , , , , . From equation (1),

Corollary 5. *Degree based on some molecular descriptors of the chemical graph of brincidofovir:*(1)*.*(2)*.*(3)*.*(4)*.*(5)*.*(6)*.*(7)*.*(8)*.*(9)*.*

*Proof. *From Table 1 and Theorem 5, the following equations are obtained:

Theorem 6. *The neighborhood M-polynomial of ** is*

*Proof. *From Figure 3, the set of edges is divided as follows: , , , , , , , ,