Computational and Theoretical Characteristics of Chemical Graph Theory and ApplicationsView this Special Issue
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.
Graph theory is the mathematical theory of the properties and applications of networks or structures that are very useful to help define and visualize relationships between various components. Basically, the graph is a composition of points of departure known as nodes and the lines formed by joining these vertices called edges . Each vertex’s degree, defined 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. The connectedness of the graphs may be used to filter and condense knowledge . The ideas and concepts of graph theory are widely used in various branches of study. The 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 . Many properties of a molecule are known to be closely linked to its graphical structure, including heat of formation, boiling temperature, potential energy, and stiffness. Researchers can use topological indices defined 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 . Chemical tests are useful to predict the power of topological indices for analyzing the physiochemical properties of molecular structures . While researching on boiling point in 1947, Wiener was the first 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 atom-bond 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.
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 denote a (molecular) graph, with and as vertex and edge sets, respectively. The number of edges incident with a vertex p is called the degree of that vertex and denoted by .
Gutman et al. [14, 15] presented the first Zagreb index and second Zagreb index almost four decades ago as
Shirdel et al.  established the “hyper-Zagreb index” as follows:
Estrada et al.  established the (atom-bond connectivity index), which is extremely important. It is described as
Vukicevic and Furtula  proposed the geometric arithmetic index , which is defined as
The Randi index, created by Randić in 1975 , was the first-degree-based topological index, which is written as
The forgotten topological index was introduced by Gutman et al.  and was defined as
Furtula et al.  offered the following improved Zagreb index:
The redefined , and Zagreb indices for the graph were established by Ranjini et al. as follows:
3. Main Results
Antimony telluride is an inorganic chemical compound with molecular formula 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 . The 2D molecular structure of antimony telluride 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 unit cells to compute antimony telluride formulae . After that, we connect the first unit cell with another unit cell in a vertical orientation, and so on up to unit cells, resulting in the antimony telluride structure shown in Figure 2.
3.1. Antimony Telluride
Theorem 1. Let represent the structure of antimony telluride for , then,
To compute vertices formulae, we now use Table 1. signifies the set of vertices of degree 3, the number of vertices of degree 5, and the number of vertices of degree 6 in Table 1.
Finally, we discovered that set of vertices of degree 3 are , set of vertices of degree 5 are , and set of vertices of degree 6 are . Using the same technique, we will split the edges to get the abstracted indices. The first edge pack has six edges , where and . The second edge pack has edges , where and . The third edge pack has edges , where and . The fourth edge pack has edges , where and . The fifth edge pack has edges , where and . In Table 2, edge partition of with is shown.(i)Atom-bond connectivity index(ii)Geometric arithmetic index The connection between and is shown in Table 3 and Figure 3(iii)Forgotten topological index(iv)Augmented Zagreb index The connection between and is shown in Table 4 and Figure 4.(v)Redefined Zagreb index(vi)Redefined Zagreb index(vii)Redefined Zagreb index The connection between , , and is shown in Table 5 and Figure 5.(viii)General Randic index For , For , For , For , The connection between the , , , and is shown in Table 6 and Figure 6.(ix)First Zagreb index(x)Second Zagreb index(xi)Hyper-Zagreb index The connection between , , and is shown in Table 7 and Figure 7.(xii)First Zagreb polynomial(xiii)Second Zagreb polynomial The connection between the and is shown in Table 8 and Figure 8.
In this study, the researcher has computed some topological indices and polynomials of the structure of antimony telluride including the atom-bond connectivity index, hyper-Zagreb index, geometric arithmetic indexes, redefined , , and Zagreb index, and Randic index of the chemical graph of antimony telluride. The numerical and visual representations of these indices are also provided. Another intriguing issue for further research is the calculation of various distance-based topological indices and the reverse topological of antimony telluride.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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