#### Abstract

A topological index is a real number derived from the structure of a chemical graph. It is helpful to determine the physicochemical and biological properties of a wide range of drugs, and it better reflects the theoretical properties of organic compounds. This is accomplished using degree-based topological indices. Vitiligo is a common, acquired skin pigmentation disorder that significantly impacts the quality of life. It frequently embodies a therapeutic challenge, resulting in interest in alternative treatments based on vitamin and herbal supplements. In this article, azathioprine, clobetasol, desonide, hydrocortisone valerate, and other drugs utilized to cure vitiligo have discoursed, and the goal of QSPR revision is to determine the mathematical relationship between properties under investigation (e.g., polarity and enthalpy) and diverse descriptors associated with the drugs’ molecule. The QSPR model will help to predict physical properties. In this study, topological indices (TIs) imposed on said drugs were found to have a good correlation with physicochemical properties in this course. Finally, this work can be helpful to design and synthesize new vitiligo treatments and other disease drugs.

#### 1. Introduction

Vitiligo is a familiar depigmenting skin disorder characterized by idiopathic, acquired, gradual, delimited hypomelanosis of the hair and skin, with a total absence of melanocytes under the microscope. Vitiligo is a serious skin disease that affects the patient’s quality of life significantly. [1]. The disease is characterized by melanocyte loss and the development of depigmented patches, which results in pigment dilution in the affected skin areas. It occurs globally, with incidence rates ranging from 0.5 to 4%, and its prevalence is comparable across genders and races [2]. Significant progress in understanding the pathogenesis of vitiligo has been made, and today, it is certainly categorized as an autoimmune disease [3]. Vitiligo ought not to be ignored as a minor or insubstantial disease, as its consequences can be psychologically catastrophic, causing profound emotional distress and, in many cases, a significantly reduced quality of daily life. Vitiligo patients may feel self-conscious or anxious about their skin. They can be rude at times, such as staring or saying hurtful things. This, in turn, may cause anxiety. Patients are most vulnerable to the disease’s negative psychosocial impact when they are between the ages of 10 and 30. It is quite often a therapeutic challenge, prompting attention in therapeutic options such as herbal and vitamin supplements. Medicos and scientists are constantly searching for more effective methods to treat vitiligo patients. One approach is to develop and test new drugs. Drug discovery is a hard process because it is expensive, time consuming, and in certain cases extremely difficult. Drugs are prescribed to treat and halt the said fatal disease, and numerous drug tests are conducted to combat the fatal disease. This necessitates prompt medical assessment, screening, and medication to assist patients in disease control. The eleven vital drugs, medicines like fluticasone propionate, clobetasone, beclomethasone dipropionate, desonide, azathioprine, clobetasol propionate, monobenzone, fluticasone, betamethasone valerate, psoralen, and hydrocortisone valerate, are safe and effective medicines that are compelled to ensure the health of the community. The chemical structure of the aforementioned drugs is depicted in Figure 1.

**(a) Fluticasone**

**(b) Clobetasone**

**(c) Beclometasone**

**(d) Desonide**

**(e) Clobetasol propionate**

**(f) Azathioprine**

**(g) Monobenzone**

**(h) Betamethasone valerate**

**(i) Psoralen**

**(j) Hydrocortisone valerate**

**(k) Fluticasone propionate**

Topological indices (TIs) are quantitative descriptors derived from a chemical graph that completely describes the chemical system and is extensively used in the research project on several drugs’ physicochemical properties. Because polynomials and TIs are widely assessed and represent the chemical structure, they play an important role in chemical graph theory. Degree-based TIs are crucially significant and play a key role in chemical graph theory. There has been a lot of interest in the use of graph invariants (TIs) in quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) studies over the last few years. TIs have applications in numerous areas of mathematics, such as bioinformatics, mathematics, informatics, and biology, but their most useful aspect to date has taken place in nonempirical QSPR [4]. Drug bioactivity can be predicted using the ABC index, Wiener index, and Randic index. The QSPR models aid in determining the most appropriate relationship between topological indices and psychochemical properties. These psychochemical properties are being examined because they all have a major impact on bioactivity and drug transport in the human body. We calculated degree-based TIs for vitiligo drugs in this paper. Likewise, vitiligo drugs are organic molecules with carefully defined topological indices that undergo purposeful QSPR analysis. The respective characteristic approximated by this method is highly correlated with the characteristic of vitiligo drugs using linear regression. There is a strong correlation between drug properties and TIs, which has been discovered.

Previous research on potential drugs against COVID-19 is discussed by Colakoglu [5]. Novel drugs used in cancer treatment were discussed by Havare [6] and discussed that drug discovery is a costly and complex phenomenon, so these are best predicted with this method. Blood cancer drug QSPR modeling [3] shows a strong correlation between TIs and drug properties. Advances in QSPR studies for various topological indices for various chemical structures motivated us to work on the current research problem. The purpose of this study is to look into the use of TIs in determining the physical properties and its QSPR modeling of vitiligo disease drug regimens used in therapeutic management.

Previous research on potential drugs against COVID-19 is discussed by Colakoglu [7]. Novel drugs used in cancer treatment were discussed by Havare [6] and discussed that drug discovery is a costly and complex phenomenon, so these are best predicted with this method. Blood cancer drug QSPR modeling was done by Nasir et al. [8] which shows a strong correlation between TIs and drug properties. Advances in QSPR studies for various topological indices for various chemical structures motivated us to work on the current research problem. The purpose of this study is to look into the use of TIs in determining the physical properties and its QSPR modeling of vitiligo disease drug regimens used in therapeutic management. Rheumatoid arthritis (RA) is a joint disease, according to Parveen et al. [9]. They used purposeful QSPR analysis and carefully crafted topological indexes to investigate the chemical components that make up RA medications. A computer method was put out by Sakander et al. [10] for computing analytically precise equations for specific degree and distance-based topological indices for generic networks. In order to demonstrate that our technique is more effective and has less algorithmic and computational complexity, some experiments are carried out in comparison to the well-known techniques. Four polynomials, Sadhana, omega, theta, and Padmakar–Ivan for double benzenoid chains, are calculated by Fozia et al. [11]. These polynomials’ analytical closed expressions are derived using the edge-cut approach. The QSPR modeling of antituberculosis drugs is detailed in [12], and Parveen et al. [13] completed the QSPR study of diabetes treatments and found a best fit model for it.

#### 2. Material and Method

In drug configuration, atoms depict vertices, and the associated bonds connecting the atoms are termed to as edges. Graph is thought to be simple, finite, and connected, whereas and in the chemical graph are referred to as vertex and the edge set, respectively. The degree of a graph vertex is the number of vertices adjacent to is denoted by . In chemistry, the valence of a compound and the degree of a vertex in a graph are concepts that are inextricably linked [4, 14–16]. We used the following degree-based topological indices:

*Definition 1. *The ABC index [17] of is defined as follows:

*Definition 2. *The first TI is Randic index introduced by Milan Randic [18]. Randic index is defined as follows:

*Definition 3. *The sum connectivity index [15] of is defined as follows:

*Definition 4. *The GA index [19] of is defined as follows:

*Definition 5. *The first and second Zagreb indices [20] of is defined as follows:

*Definition 6. *The harmonic index [21] of is defined as follows:

*Definition 7. *The hyper Zagreb index [22] of is defined as follows:

*Definition 8. *The forgotten index [23] of is defined as follows:

The -electron energy of a molecule was calculated using the first and second Zagreb indices [16]. The heat of the creation of alkanes can be preeminently predicted using the augmented Zagreb index [24]. ChemSpider is used to calculate the values of physical properties.

Table 1 shows that the data is normally distributed. As a result, the linear regression model is best to check and use for the aforementioned analysis. We endorse that readers read the following articles [3, 6, 14, 24–26]. Monobenzone propionate has a molecular formula of C_{13}H_{12}O_{2}. It is a hydroquinone derivative that is used in the treatment of vitiligo. It is the monobenzone ether of hydroquinone, which is used in medicine to treat pigmentation. This medication comes in the form of a white, nearly tasteless white crystalline that is soluble in organic solvents but practically insoluble in water. It has a depigmenting effect on mammalian skin by increasing melanin excretion from melanocytes. It may also cause melanocyte destruction and permanent depigmentation. Monobenzone works by effectively removing colour from normal skin around vitiligo skin. Fluticasone has the molecular formula of C_{22}H_{27}F_{3}O_{4}S. It cures corticosteroid-responsive dermatoses. Clobetasone has the formula of C_{22}H_{26}ClFO_{4}. It is frequently used topically as a treatment for a variety of ailments. It is often employed topically as a treatment for a variety of conditions such as eczema, various forms of dermatitis, psoriasis, and for certain ophthalmologic conditions. When cortisol derivatives are applied to the skin, they produce anti-inflammatory, antiproliferative, immunosuppressive, and vasoconstrictor effects. Topical clobetasone butyrate is used in dermatology to heal itchiness and erythema caused by eczema and dermatitis. Clobetasone and its metabolites are eliminated through the urine. Beclomethasone dipropionate has the molecular formula of C_{28}H_{37}N_{7}ClO_{7}. In 1972, it was first available in a pressurized metered-dose inhaler, followed by a dry powder inhaler and an aqueous nasal spray. Beclomethasone dipropionate is used to treat inflammatory conditions such as asthma, dermatoses, and allergic rhinitis because of its anti-inflammatory, antipruritic, and antiallergy properties and excreted in urine. Desonide has a molecular formula of C_{24}H_{32}O_{6}. It is a nonfluorinated synthetic corticosteroid used topically in dermatology.

Corticosteroids are a group of steroids and used as anti-inflammatory and antipruritic agents. Betamethasone is used to relieve inflammation in several conditions such as an allergic and dermatologic disorder. It topically manages inflammatory skin conditions including autoimmune disorder. Clobetasol propionate has the molecular formula of C_{25}H_{32}ClFO_{5}. It is a corticosteroid that is used to treat corticosteroid-responsive dermatomes as well as plaque psoriasis. Azathioprine propionate has the molecular formula of C_{9}H_{7}N_{7}O_{2}S. It is an immunosuppressant that is helpful to reduce Crohn’s disease, rheumatoid arthritis, and ulcerative colitis and also to prevent renal transplant rejection. It is used to treat inflammatory diseases such as rheumatoid arthritis. Hydrocortisone valerate has the molecular formula of C_{26}H_{38}O_{6}. It is a corticosteroid that is used to treat pruritic dermatoses and inflammation that are responsive to corticosteroids. It is also employed in the treatment of endocrine (hormonal) disorders. It is also used to treat a variety of allergic and immune conditions, including severe asthma, severe psoriasis, arthritis, and lupus. Psoralen is the parent chemical substance in a group of organic compounds in nature that are employed to heal vitiligo. Fluticasone propionate has the molecular formula of C_{25}H_{31}F_{3}O_{5}S. This is a glucocorticoid medication that is used to treat asthma, inflammatory pruritic dermatoses, and nonallergic rhinitis.

#### 3. Results and Discussions

In this section, degree-based TIs are executed on vitiligo drugs. The relation between QSPR analysis and topological indices portrays that the properties are vastly correlated in terms of physicochemical properties for the said disease. The eleven medicines, fluticasone propionate, clobetasone, beclomethasone dipropionate, desonide, azathioprine, clobetasol propionate, monobenzone, fluticasone, betamethasone valerate, psoralen, and hydrocortisone valerate, are used in the analysis for vitiligo. The drug structures are displayed in Figure 1. We consider the molecular structure as graph, and the drug elements denote vertices and bonds among atoms are their edges. We use regression analysis calculation for drug study.

##### 3.1. Regression Model

In this article, drug computable structure analysis of nine topological indices for QSPR modeling tenacity is performed. The five physical properties, refractivity (R), polarity, complexity, molar volume (MV), and enthalpy (E) for eleven medicines used in vitiligo treatment, are listed in Table 2. We conduct the regression analysis for the drugs, and the linear regression model is tested using the following equation: where denotes the physicochemical property of the given drug. The stands for topological index, stands for constant, and stands for regression coefficient. The Statistix, SageMath, and MATLAB software are useful for determining the results. A linear QSPR model is used to analyze nine TIs of vitiligo drugs and their physiochemical properties. Equation (9) is pertinent for the aforementioned calculation.

Theorem 9. *Let be the graph Psoralen. The various TIs of are given as follows:
*

*Proof. *Let be graph of psoralen and let represent the class of edges of joining vertices with and .
(i)By using Definition 1, we get the following:(ii)By using Definition 2, we get the following:(iii)By using Definition 3, we get the following:(iv)By using Definition 4, we get the following:(v)By using Definition 5, we get the following:(vi)By using Definition 5 and above given edge partitions , we get the following:(vii)By using Definition 6, we get the following:(viii)By using Definition 7, we get the following:(ix)By using Definition 8, we get the following:

Theorem 10. *Let be graph of azathioprine. The various topological indices of are given as follows:
*

*Proof. *Let be the graph of azathioprine and let represent the class of edges of joining vertices with and .
(i)By using Definition 1, we get the following:(ii)By using Definition 2, we get the following:(iii)By using Definition 3, we get the following:(iv)By using Definition 4, we get the following:(v)By using Definition 5, we get the following:(vi)By using Definition 5, we get the following:(vii)By using Definition 6, we get the following:(viii)By using Definition 7, we get the following:(ix)By using Definition 8, we get the following:

Topological indices for the remaining drugs can be calculated using the same procedure as in Theorems 9 and 10 and Definitions 1–8. To reduce the length of paper, only two drug calculations are added. Table 1 also includes the calculated values for all drugs’ TIs. Figure 2 depicts a graphical representation of calculated TIs for various medicines. Using Equation (9), we calculated the getting-ready linear models for all TIs, which are listed as follows: (1)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity =(2)Regression models for :

Enthalpy=

Polarity=

Molar volume=

Complexity=

Refractivity=(3)Regression models for :

Enthalpy=

Polarity=

Molar volume=

Complexity=

Refractivity=(4)Regression models for :

Enthalpy=

Polarity=

Molar volume=

Complexity=

Refractivity=(5)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity=(6)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity=(7)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity=(8)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity=(9)Regression models for :

Enthalpy =

Polarity=

Molar volume=

Complexity=

Refractivity=

Tables 3–11 represent the statistical parameters used in QSPR models of TIs.

##### 3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation Coefficient of Physicochemical Properties

Table 2 shows physical properties of eleven vitiligo drugs, and Table 1 shows TIs computed using molecular structure. Table 12 lists correlation coefficients between five physical properties and TIs. Figure 3 depicts the graph between TIs and physical properties.

**(a) Complexity on TI**

**(b) Enthalpy on TI**

**(c) Molar volume on TI**

**(d) Polarity on TI**

**(e) Refractivity on TI**

##### 3.3. Calculation of Statistical Parameters

In this section, we find the relation between degree-based TIs and physical properties of vitilgo drugs such as medicines fluticasone propionate, clobetasone, beclomethasone dipropionate, desonide, azathioprine, clobetasol propionate, monobenzone, fluticasone, betamethasone valerate, psoralen, and hydrocortisone valerate, and this is achieved through the use of QSPR modeling. TIs, , , and represent the independent variable, regression model constant, correlation coefficient, and sample size, respectively. The said kind of test can be useful for comparing and deciding on model improvements. It is worth noting that is higher than 0.6 and thevalues are almost higher than 0.05. As a result, it determines that all properties are significant.

##### 3.4. Standard Error of Estimate (SE), Correlation Determination, and Comparison

The standard error estimate is the measure of variation for an observation calculated around the computed regression line. It examines extent of correctness of predictions made about the calculated regression line in Table 13. In Table 14, the percentage of relationship described by correlation determination gives ample information about the relationship between variables. It is calculated by squaring the value of . Tables 15–19 compare the experimental and theoretical measurement results of the models’ physicochemical properties.

#### 4. Conclusions

The statistical parameters used during linear QSPR models and TIs demonstrate that index provides high correlated value for molar volume . index offers maximum correlated value of complexity, i.e., . index depicts utmost correlation coefficient of refractivity . Harmonic provides maximum correlated value of enthalpy .

The QSPR modeling is crucial because it makes physical properties more predictable. It offers a technique to do away with time-consuming experimenting and saves time. Without conducting any experiments, the elusive are anticipated. QSPR modeling is beneficial to create and forecast drug characteristics. This technique will be used to forecast in addition to create novel drugs for the future treatment of additional ailments. Getting creation of drugs is not a simple task because it may be expensive, time consuming, and difficult at times. But this approach is superior and efficient in producing the need. In this paper, we calculated TIs and compared them to a linear QSPR model for drugs used to treat vitiligo. The findings acquired in this manner would be useful in the pharmaceutical industry in inventing better drugs to obtain precautionary measures for the aforementioned disease. The correlation coefficient makes a significant contribution to the scope of TIs for such drugs. The observations are eye opening for pharmaceutical researchers working on drug science, and they offer a method to predict physicochemical properties for amateur inventions of many other specific diseases.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.