Ve-Degree, Ev-Degree, and Degree-Based Topological Indices of Fenofibrate

e molecular topology of a graph is described by topological indices, which are numerical measures. In theoretical chemistry, topological indices are numerical quantities that are used to represent the molecular topology of networks. ese topological indices can be used to calculate several physical and chemical properties of chemical compounds, such as boiling point, entropy, heat generation, and vaporization enthalpy. Graph theory comes in handy when looking at the link between certain topological indices of some derived graphs. In the ongoing research, we determine ve-degree, ev-degree, and degree-based (D-based) topological indices of fenobrate’s chemical structure. ese topological indices are the Zagreb index, general Randić index, modied Zagreb index, and forgotten topological index. ese indices are very helpful to study the characterization of the given structure.


Introduction
Graph theory is a mathematical sub eld. It begins in 1736 with the Konigsberg bridge problem when Leonhard Euler publish his work on the seven bridges of the city Konigsberg. Graph theory is widely used in the study of networks, models, circuits, routes, etc.. is theory has found signi cant applications in computing, chemistry, physics, electricity and civil engineering, communication science, operations research, architecture, genetics, sociology, psychology, anthropology, linguistics, and economics. It is mostly used in organic chemistry and is called chemistry graph theory. Graph theory is based on vertices and edges; in chemical graph theory, vertices represent items and edges represent bonding. In graph theory, we understand the terms of quantitative structure-property relation (QSAR) and quantitative structure-activity relation (QSPR) for structure or formula of compound.
Topological indices are numerical values associated with a chemical structure and are used to correlate a chemical structure with many properties such as chemical stability, pharmacological activity, and physical properties. ey are developed from molecular topology modeling and are divided into explicit structural descriptions such as chemical quantum and hidden structural descriptors such as hydrophobicity and electrical constants. e class of feno brate is referred to as antilipemic agents. It increases the natural processes to remove bad cholesterol from body. It decreases the fatty substances which are bad and also helps the fatty substances which are good for human body and reduces the risk of heart disease. Cholesterol has bene ts when it remains in limit level, but when it increases from its level, it becomes a danger for the heart. In cholesterol, there are three main fats, low-density lipoprotein (LDL) and triglyceride are bad when increased but highdensity lipoprotein (HDL) is known as good cholesterol.
Fenofibrate is used to reduce LDL and triglyceride levels along with a low-fat diet and exercise. e main purpose of our research is to control the heart disease with the help of fenofibrate using topological index. We calculate the formula via ve-degree, ev-degree, and degree-based topological indices. In chemical graph theory, vertices represent atoms and edges represent bonding. e graph of a chemical structure is a simple undirected graph. A graph is a combination of nonempty sets ℷ and Ξ. e members are vertices and edges, respectively. δ(ℸ) is the degree of vertex ℸ and is the number of edges incident to ℸ. e set of vertices which are connected to ℸ is open neighborhood of ℸ, denoted by N(ℸ). N(ℸ) + ℸ are the closed neighborhoods of ℸ. A topological index is a unique number that defines the underlying structure of the molecular graph.
ese topological indices are used to study the quantitative structure-property and relationships. Scientists use the numerical parameters which are obtained from chemical graph network in QSPR and QSAR. In drug designs, its applications are in. e concept of topological index was firstly introduced by Wiener, and it was distance-based topological index. Later in 1972-75, D-based Zagreb index and Randić index were introduced. Recently, Chellali et al. [4] proposed two new degree concepts in topological indices, called ve-degree and ev-degree. Math concepts are discussed in [5]. e classical degree concepts (see [6][7][8]) have been converted to ve-degree and ev-degree such as Zagreb index, Randić index, modified Zagreb index, inverse degree index, and forgotten index (see [9][10][11]). It was found that they give better prediction results than classical indices [12,13].

Motivation.
In the last two years, COVID-19 has surged dramatically in numerous nations. Many pharmacists produced a variety of medications to aid in the management of COVID-19. Fenofibrate is a substance that is found in various anti-COVID-19 medications. In laboratories, studying its physicochemical properties is highly expensive and time consuming. We work on it and derive the mathematical equation utilising topological indices that are useful to chemists in order to improve it.

Preliminaries
Here we give basic concepts. Consider a graph Δ with ℷ vertex set and Ξ edge set. e degree is the number of edges incident to vertex ℸ, denoted as δ(ℸ).
e ve-degree of vertex ℸ is the number of edges which are incident to any vertex of close neighborhood of ℸ, denoted by δ ve (ℸ). e number of vertices in the union of closed neighborhoods of ℸ and ℷ is the ev-degree of edge e � ℸℷ, denoted by δ ev (e).
e first Zagreb index [14] introduced in 1972 is defined as e ve-degree Zagreb beta index of graph Δ is defined as (2) e ve-degree Zagreb alpha index of Δ is defined as e ev-degree Zagreb index is defined as e Randić index [15] is defined as e ve-degree Randić index of graph Δ is defined as e ev-degree Randić index is defined as e ve-degree modified Zagreb index is defined as e F index [16] is defined as e ve-degree F index is defined as e ev-degree F index of graph Δ is defined as

Applications of Topological Indices.
Wiener created the idea of topology index while investigating alkane boiling points. Topological indices are used to studies the quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) to compare the topological similarities of different chemical compounds. Randić index has been closely correlated with many chemical properties, but the GA index is better than Randić connectivity index. Topological indices encode information regarding molecular size, shape, branching, and so on in numerical form, which is used for measuring topological similarity between chemical compounds and in (QSPR)/ (QSAR) studies. Randić index has been closely correlated with many chemical properties and found to parallel the boiling point and Kovats constants. For certain physicochemical properties, the predictive power of GA index is somewhat better than predictive power of the Randić connectivity index.

Methods
To calculate results, we use the vertex segment technique, edge partition technique, degree checking technique, and degree of neighbors strategy. We use Maple 16 for verification of calculations.

Fenofibrate
Structure. e 2D molecular graph of fenofibrate is given in Figure 1.  Table 1 shows the edge partition of fenofibrate. e molecular graph of fenofibrate is shown in Figure 2.

D-Based Zagreb Index.
We calculate degree-based Zagreb index using Table 1.

Ev-D-Based Zagreb Index.
We calculate ev-degree Zagreb index using Table 2. Table 3, we calculate ve-degree Zagreb alpha indices. Zagreb alpha index:

Conclusion
Topological descriptors have an important role in the study of QSPR/QSAR and predict the properties of chemical compounds. Fenofibrate is a drug that is used in conjunction with a healthy diet to lower cholesterol and triglyceride (fatlike substance) levels in the blood. is could help prevent pancreatitis (inflammation or swelling of the pancreas) caused by excessive triglyceride levels in the blood. In this study, we compute D-based first Zagreb index, ve-degree Zagreb α index, Zagreb β index, ev-degree Zagreb index, Randić index, ve-degree Randić index, ev-degree Randić index, modified Zagreb index, ve-degree modified Zagreb index, ev-degree modified Zagreb index, F index, ve-degree F index, and ev-degree F index for fenofibrate. With the help of these, we can study the physicochemical properties of the fenofibrate. Chemical compound testing in a lab is very expensive and time-consuming, whereas TIs are simple to obtain. e production of the drug fenofibrate can be obtained in very low cost, which can be frequently used everywhere. Future work may be done to investigate TI for other drugs.

Data Availability
No data were used to support this study.