Computing Topological Indices and Polynomials of the Rhenium Trioxide

In the study of mathematical chemistry and chemical graph theory, a topological index, also known as a connectivity index, is the arithmetical framework of a graph that speci es its topology and also graph invariant. ­ese topological indices are used to model quantitative structure relationships (QSARs), which are connections between the work of biological or other molecular structures and the chemical structures. ­is study computed the rst, second, and Hyper Zagreb indices, as well as Zagreb polynomials, Rede ned Zagreb indices, Randic index, ABC index, and GA index of chemical structure of Rhenium Trioxide.


Introduction
Graph theory is the branch of mathematics dealing with graphs which are basically the structures used to model pairwise relationship in di erent things. A graph consists of vertices that are a xed to edges [1]. A graph G is formed of V(G) ≠ ϕ of elements known as vertices, having nite numbers E(G) of unorganized pairings for V(G) units known as edges [2]. V(G) and E(G) were assigned to the vertex and edge sets of G, respectively. In chemical graph theory, atoms are represented by vertices and bonds are represented by edges [3,4]. Chemical graph theory (GT) is the discipline of mathematical chemistry where we quantitatively implement devices of the graph premise to a material [5,6]. Now a days, the proposal makes a signi cant contribution to computational sciences. As a result, we can de ne a subatomic diagram as a constrained network in which the nodes representing atoms and edges represent covalent bonds in concealed complex formation [7]. Symmetric key is a analytical standard associated to the composition that indicates the relationship between the concentration of the element and a wide range of physiosynthetic qualities, composite sensitivity, and organic action [8]. Molecular graphs are all chemical structures having structural formulas that comprise covalently bond compounds or molecules. Chemical graph theory tools were previously used for a variety of reasons, including numeration, systemization of the topic being discussed, and nomenclature [9,10]. It also describes the process of assembling rules or regulations including a framework or strategy, as well as computer programming [11,12]. e existence of isomerism, that is justi ed by the component graph hypothesis reinforces the necessity of graph theory techniques of science [13,14]. e computational geometry of molecules according to well-de ned rules is the essence of chemistry [15]. A topological graph index is a mathematical formula that may be applied to any graph that describes a molecule structure. e corresponding counting corresponding polynomials are some most common topological indices [16,17]. On the other way, topological indices (TIs) are analytical principles related to chemical compositions for determining the relationship between chemical structure and various attributes. Physical qualities such as boiling temperatures, molar heats of formation, thermodynamic data, as well as chemical reactivity such as octane values as well as reactivity data, and biological activity are among these features [8,18]. Topological indices, in particular, appear to hold promise for last mentioned goal, specifically QSAR [19,20]. e quantitative structure-activity relationship (QSAR) is a computational modelling technique for determining connections between chemical compound structural features and biological activities [21]. In this research, we discuss about the inorganic compound ReO 3 rhenium trioxide or rhenium(VI) oxide. It is a bright red solid with a metallic lustre that looks like copper. It is the Group 7 th element, only stable trioxide. Having average estimation absorption of 1 part per billion, the most infrequent element on Earth crust is rhenium. From any stable elements, rhenium trioxide is the only element that has the 3 rd highest melting point of 5903K. In this research, we talk about different topological indices and polynomials of rhenium trioxide.

Preliminaries
In the world of chemical graph theory, wiener index (WI) which is known as wiener number presented by Henry wiener [22]. Actually, wiener index is the topological index of a molecule, stated nonhydrogen atoms in a molecule are denoted by total lengths of the shortest path in all pairs of vertices in a chemical graph. e number of edges linking a vertex determines its degree in graph theory. If ′ r ′ and ′ r ′ are the vertices of a graph, then Ψ(t), Ψ(r) are the degree of ′ t ′ and ′ r ′ vertices, respectively.
Gutman and Trinajstic introduced in his study 1 st , 2 nd Zegreb indices as (1) Shirdel et al. established "hyper Zagreb index" as follows: Atom bond connectivity index is the most important and well known connectivity topological indices, presented by Estrada et al.: "Geometrically arithmetic index (GA)" is another important topological figure of connectivity which is presented by Vukicevic et al.: Gutman and Trinajstic presented 1 st and 2 nd Zagreb polynomial as Redefined 1 st , 2 nd , and 3 rd Zagreb indices are the most significant indices presented by Usha et al. stated below as , ere is also well known topological index called Randic in 1975 presented as SCI was introduced by Zhou et al. and defined as

Main Results of Rhenium Trioxide
Rhenium is the derived from the Latin word Rhenus which means "Rhine" is the second last element that is discovered and having stable isotope. Rhenium is a chemical element that is denoted by ′ Re ′ with atomic number 75. It is silver gray colored, heavy, and 3 rd row transition metal in 7 th group of periodic table. Rhenium look like manganese and technetium chemically in nature; it is acquired by the consequence of ancestry and purification copper ores and molybdenum. Rhenium expresses the number of oxidation states in the wide limitation of −1 to +7.As compared others, the oxide ReO 3 is the best electrical conductor at room temperature resistively that is close to copper, as shown in Figure 1. e cubic crystal of ReO 3 identifies particular perovskite-type crystal structure that is related to O h 1 (Pm3m) having lattice parameter a 0 � 3.7504A 0 . e crystal lattice of ReO 6 contains octahedra joined from edges and the unit cell of Bravais that holds f.u. of ReO 3 . e red colored rhenium trioxide shows metallic conductivity, it is also called covalent metal (see Figure 1).

Theorem 1.
If R is the graph of rhenium trioxide as illustrated in Figure 1, then  Proof. Let G be isomorphic to rhenium trioxide. en, the edge partition for the chemical graph of rhenium trioxide based on degree sum of vertices of each other is given by □

Hyper Zagreb Index
e comparison between M 1 (G), M 2 (G), and HM(G) is shown in Table 1 and Figure 2.

Second Zagreb Polynomial
e comparison between first and second Zagreb polynomials is represented in Table 2 and Figure 3.

Redefined First Zagreb Index
Journal of Mathematics   Table 1.     Table 2.

Redefined ird Zagreb Index
Journal of Mathematics e comparison between redefined Zagreb indices is represented in Table 3 and Figure 4.

Harmonic Index
e connection between the RI(R), SCI(R), and HI(R) is shown in Table 4 and Figure 5.

Journal of Mathematics
e connection between ABC(R) and GA(R) is shown in Table 5 and Figure 6

Conclusion
In this research work, researchers computed different topological indices such as Zagreb indices, redefined Zagreb indices, atom bond connectivity, harmonic index and sum connectivity index, and many other indices of chemical graph of rhenium trioxide (ReO 3 ). Rhenium trioxide also has been expanded by a researcher upto m and n cycles. Researchers also explained the comparison between the different versions of topological indices in a numerical way with the help of table and also expressed them in the graphical pattern. By comparing these topological indices with tables and graphs, the importance of indices shows that these indices results have significant relationship with each other.

Data Availability
No data were used to supoort this study.

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