Topological Descriptors and Polynomials for Analysing the Structure of Antimony Telluride

In a molecular network, molecules are correlated with numerical values, which are referred to as topological indices. The chemical and physical properties of chemical substances can be determined using topological indices. Mathematicians frequently use topological indices to calculate the strain energy, melting point, boiling temperature, distortion, and stability of chemical compounds. Topological indices also serve as a connection between a compound’s biological activity and its physical properties. In this research paper, we have computed degree-based topological descriptors and polynomials to analyze the structure of antimony telluride.


Introduction
Graph theory is the mathematical theory of the properties and applications of networks or structures that are very useful to help de ne and visualize relationships between various components. Basically, the graph is a composition of points of departure known as nodes V(G) and the lines formed by joining these vertices called edges E(G) [1]. Each vertex's degree, de ned as the number of edges that enter or leave it, is a key parameter. Graphs serve as a model for a broad range of systems whose function and structure are determined by the component elements' connection model. e connectedness of the graphs may be used to lter and condense knowledge [2]. e ideas and concepts of graph theory are widely used in various branches of study. e major role of graph theory in chemical graphs is the mathematical modeling of chemical phenomena. We can generalize the model of a molecule where each vertex represents an atom, and the edges connecting the relevant vertices indicate covalent connections between atoms [3,4]. Chemists can easily gather details buried in the balance of the subatomic diagram and anticipate attributes of compounds by assigning numbers and polynomials to chemical compounds. Topological indices have a wide range of applications in theoretical chemistry, particularly in QSPR/ QSAR research [5]. Many properties of a molecule are known to be closely linked to its graphical structure, including heat of formation, boiling temperature, potential energy, and sti ness. Researchers can use topological indices de ned on chemical molecule architectures to better understand physical features, chemical reactivity, and biological activity [6,7].
In the current era, research in degree-based topological indices has been rising thoroughly. In recent years, several research papers have been published considering topological indices [8]. Chemical tests are useful to predict the power of topological indices for analyzing the physiochemical properties of molecular structures [9]. While researching on boiling point in 1947, Wiener was the rst to develop the notion of the topological index in chemical graph theory [10,11]. In this research, the structure of antimony telluride is formulated in which a single atom of antimony makes a bond with six atoms of tellurium, and every tellurium makes a bond with three antimony atoms and three tellurium atoms that make a compound crystallizing in a hexagonal lattice it creates a closed packed structure. We have computed the atombond connectivity index, hyper-Zagreb index, geometric arithmetic indexes, redefined first, second, and third Zagreb index, and Randic index of the chemical graph of antimony telluride.

Preliminaries
Topological indices are graph invariant, which means they remain the same regardless of how the graph is constructed or named. Giving chemical compound numbers and polynomials makes it easier to explain their physical characteristics [12,13]. Let G denote a (molecular) graph, with V(G) and E(G) as vertex and edge sets, respectively. e number of edges incident with a vertex p is called the degree of that vertex and denoted by Γ(p).
Gutman et al. [14,15] presented the first Zagreb index (M 1 ) and second Zagreb index (M 2 ) almost four decades ago as Shirdel et al. [16] established the "hyper-Zagreb index" as follows: Estrada et al. [17] established the ABC (atom-bond connectivity index), which is extremely important. It is described as Vukicevic and Furtula [18] proposed the geometric arithmetic index GA, which is defined as e Randi index, created by Randić in 1975 [19], was the first-degree-based topological index, which is written as e forgotten topological index was introduced by Gutman et al. [20] and was defined as Furtula et al. [21] offered the following improved Zagreb index: e redefined 1st, 2nd, and 3rd Zagreb indices for the graph G were established by Ranjini et al. [22]as follows: ,

Main Results
Antimony telluride is an inorganic chemical compound with molecular formula (Sb 2 Te 3 ) in which a single atom of antimony makes a bond with six atoms of tellurium, and every tellurium makes a bond with three antimony atoms and three tellurium atoms that make a compound crystallizing in a hexagonal lattice it creates a closed packed structure [23]. e 2D molecular structure of antimony telluride (Sb 2 Te 3 ) is given in the figure below. We used the following configuration to characterize its molecular graph: the number of unit cells in a row that is linked is denoted by "m", whereas the total number of linked rows with m cells is denoted by "n". In Figure 1, we showed how cells in a row connect to one another and how one row relates to another.
For the calculation of antimony telluride formulae, we utilize a unit cell initially and then combine it with another unit cell in a horizontal manner, and so on up to m unit cells to compute antimony telluride formulae [23]. After that, we connect the first unit cell with another unit cell in a vertical orientation, and so on up to n unit cells, resulting in the antimony telluride structure shown in Figure 2.  To compute vertices formulae, we now use Table 1. V 3 signifies the set of vertices of degree 3, V 5 the number of vertices of degree 5, and V 6 the number of vertices of degree 6 in Table 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.