Some Novel Results Involving Prototypical Computation of Zagreb Polynomials and Indices for SiO4 Embedded in a Chain of Silicates

A topological index as a graph parameter was obtained mathematically from the graph’s topological structure. These indices are useful for measuring the various chemical characteristics of chemical compounds in the chemical graph theory. The number of atoms that surround an atom in the molecular structure of a chemical compound determines its valency. A significant number of valency-based molecular invariants have been proposed, which connect various physicochemical aspects of chemical compounds, such as vapour pressure, stability, elastic energy, and numerous others. Molecules are linked with numerical values in a molecular network, and topological indices are a term for these values. In theoretical chemistry, topological indices are frequently used to simulate the physicochemical characteristics of chemical molecules. Zagreb indices are commonly employed by mathematicians to determine the strain energy, melting point, boiling temperature, distortion, and stability of a chemical compound. The purpose of this study is to look at valency-based molecular invariants for SiO4 embedded in a silicate chain under various conditions. To obtain the outcomes, the approach of atom–bond partitioning according to atom valences was applied by using the application of spectral graph theory, and we obtained different tables of atom—bond partitions of SiO4. We obtained exact values of valency-based molecular invariants, notably the first Zagreb, the second Zagreb, the hyper-Zagreb, the modified Zagreb, the enhanced Zagreb, and the redefined Zagreb (first, second, and third). We also provide a graphical depiction of the results that explains the reliance of topological indices on the specified polynomial structure parameters.


Introduction
A molecular structure is defined as a simple and linked network G, where |G| is the set of atoms (nodes) and V G is the set of atom-bonds (links between atoms) [1]. If two atomṡ a 1 andȧ 2 form an atom-bond in G, we writeȧ 1 ∼ȧ 2 ; similarly, if two atoms do not form an atom-bond in G, we writeȧ 1 ȧ 2 . The topological index of a chemical composition is a numerical value or a continuation of a given structure under discussion, which indicates chemical, physical, and biological properties of a chemical molecule, see for details [2][3][4]. Topological indices and polynomials capture molecular structural symmetries and provide mathematical vocabulary for predicting features, such as boiling temperatures, viscosity, radius of gyrations, and so on [5,6].
Mathematical chemistry describes how to use polynomials and functions to offer instructions concealed in the symmetry of molecular graphs, and the graph theory has many applications in modern chemistry, particularly organic chemistry. In chemical graph theory, the atoms and bonds of a molecular structure are represented by vertices and edges, respectively [7]. Many applications of topological indices are employed in theoretical chemistry, particularly in research pertaining to quantitative structure-property relationships (QSPRs) and quantitative structure-activity relationships (QSARs) [8][9][10]. Many famous researchers have studied topological indices to obtain information about different families of graphs [11,12]. In (QSPR) and (QSAR), topological indices are utilized directly as simple numerical descriptors in comparison with physical, biological, and chemical characteristics of molecules, which are benefits. Many researchers have worked on various chemical compounds and computed topological descriptors of various molecular graphs during the last few decades [13,14].
The molecular graph is a simple connected graph in a chemical graph theory that contains chemical atoms and bonds, which are often referred to as vertices and edges, respectively, and there must be a linkage between the vertex set V G and edge set E G . The valency of each atom of G is actually the total number of atoms connected to v of G and is denoted by d v , [15].
In 1972, Gutman and Trinajstic initiated the idea of computing the branching of the carbon-atom skeleton, which was, later on, known as the first Zagreb index [16]. In 2004, Gutman and Das, adulated characteristics of the first and second Zagreb polynomials for chemical graphs of a chemical compound, which we studied in the research articles [17]. The first Zagreb polynomial corresponding to the first Zagreb index is defined as The second Zagreb polynomial, which corresponds to the second Zagreb index [17], is written as In 2013, Shirdel et al. initiated the concept of the hyper-Zagreb index [18]. The hyper-Zagreb polynomial and index are defined as follows: The modified Zagreb polynomial and index [19] are defined as In 2010, Furtula et al. introduced the augmented Zagreb index [20]. The augmented Zagreb polynomial and index are defined as In 2013, Ranjini, Lokesha, and Usha presented [21] a redesigned version of the Zagreb indices ReZG1, ReZG2, and ReZG3. The indices and redefined form of the Zagreb polynomial are as follows: (8) In this article, the above-defined eight Zagreb polynomials and Zagreb indices were constructed by the atom-bond set of silicates, partitioned according to the valencies of the S i and O 2 atoms, [22]. We also investigate silicon tetrahedron S i O 4 in a compound structure and derived the precise formulas of certain essential valency-based Zagreb indices using the approach of the atom-bond partitioning of the molecular structure of silicates; for details, see [23,24].

Chain of Silicates
The basic unit of silicates is a SiO 4 tetrahedron, which is obtained by metal carbonates with sand or fusing metal oxides [25]. Almost all of the silicates contain SiO 4 tetrahedron. From a chemical point of view, for a tetrahedron SiO 4 , we consider a pyramid with a triangular base (single tetrahedron SiO 4 ), as shown in Figure 1, containing oxygen atoms O 2 at the four corners of the tetrahedron, and the silicon atom S i is bonded with equally spaced atoms of O 2 . From the resulting SiO 4 , a silicate tetrahedron joins with other SiO 4 horizontally, and a single chain of silicates is obtained. Similarly, when two molecules of SiO 4 join corner-to-corner, then each SiO 4 shares its O 2 atom with the other SiO 4 molecule, as seen in Figure 1. After completing this process of sharing, these two molecules of SiO 4 can be joined with two other molecules. Now, we obtain a chain of silicates SC p q , where p and q are the silicate chain numbers formed and the total number of SiO 4 in one silicate chain, respectively. Here, in the chain of silicates SC p q , pq is the number of tetrahedron SiO 4 used, see Figure 1.
Here, in the chain of silicates SC  Table 1 provides the partition of the set of atom-bonds. Table 1. Atom-bond partition of SC p q for p = q.
Proof. Using the atom-bond partition from Table 1, in the formula of the first Zagreb polynomial (1), we have This gives By taking the first derivative of the polynomial in Theorem 1 at y = 1, we obtain the first Zagreb index of the silicate network SC p p as follows: For p > 1, the first Zagreb index of SC p p is 63p 2 − 27p.
Proof. Using the atom-bond partition from Table 1, in the formula of the second Zagreb polynomial (2), we have By taking the first derivative of the polynomial in Theorem 2 at y = 1, we obtain the second Zagreb index of the chain of silicates SC p p as follows: For p > 1, the second Zagreb index of SC p p is 162p 2 − 1135p + 18.
Proof. Using the atom-bond partition from Table 1, in the formula of the hyper-Zagreb polynomial (3), we have By taking the first derivative of the polynomial in Theorem 3 at y = 1, we obtain the hyper-Zagreb index of the chain of silicates SC p p as follows: For p > 1, the hyper-Zagreb index of SC p p is 675p 2 + 513p + 36. Proof. Using the atom-bond partition from Table 1, in the formula of the modified Zagreb polynomial (4), we have Proof. Using the atom-bond partition from Table 1, in the formula of the first redefined Zagreb polynomial (6), we have This gives By taking the first derivative of the polynomial in Theorem 6 at y = 1, we obtain the first redefined Zagreb index of the chain of silicates SC p p as follows: For p > 1, the first redefined Zagreb index of SC p p is 5 2 p 2 + 2p − 1 2 .
Proof. Using the atom-bond partition from Table 1, in the formula of the second redefined Zagreb polynomial (7), we obtain This gives By taking the first derivative of the polynomial in Theorem 7 at y = 1, we obtain the second redefined Zagreb index of the chain of silicates SC p p as follows: For p > 1, the second redefined Zagreb index of SC p p is 15p 2 − 34 3 p + 1.
Proof. Using the atom-bond partition from Table 1, in the formula of the third redefined Zagreb polynomial (8), we obtain
By taking the first derivative of the polynomial in Theorem 8 at y = 1, we obtain the third redefined Zagreb index of the chain of silicates SC p p as follows: For p > 1, the third redefined Zagreb index of SC p p is 188p 2 − 1842p + 188.

Comparison
In this section, we present a numerical comparison of Zagreb indices in Table 2 and graphical comparison in Figure 2 of Zagreb polynomials for p, q > 1 and p = q = 2, 3, 4, ..., 12 for the chain of silicates SC p q .

Zagreb Polynomials and Indices for p < q and p are Odd
Here, in the chain of silicates SC p q , we observed for p < q that p is odd and the atombond on the basis of the valency of every atom of SC p q changed. So, on the basis of valency, Table 3 provides the partition of the set of atom-bonds.
By taking the first derivative of the polynomial in Theorem 9 at y = 1, we obtain the first Zagreb index of the silicate network SC p q as follows: Let p be odd and p < q. Then the first Zagreb index of SC p q is 63pq − 216p − 6q − 3.
By taking the first derivative of the polynomial in Theorem 10 at y = 1, we obtain the second Zagreb index of the chain of silicates SC p q as follows: Let p be odd and p < q. Then the second Zagreb index of SC p q is 162pq − 99p − 36q + 9.
By taking the first derivative of the polynomial in Theorem 11 at y = 1, we obtain the hyper-Zagreb index of the chain of silicates SC p q as follows: Let p be odd and p < q. Then the hyper-Zagreb index of SC p q is 675pq − 387p − 162q − 9.
Theorem 12. Let p be odd and p < q. Then the modified Zagreb polynomial of SC p q is 3(p + 1)y 1 9 +(3pq + p + 2q − 5)y 1 18 Proof. Using the atom-bond partition from Table 3, in the formula of the modified Zagreb polynomial (4), we obtain Proof. Using the atom-bond partition from Table 3, in the formula of the augmented Zagreb polynomial (5), we obtain Theorem 14. Let p be odd and p < q. Then the first redefined Zagreb polynomial of SC p q is 3(p + 1)y Proof. Using the atom-bond partition from Table 3, in the formula of the first redefined Zagreb polynomial (6), we obtain This gives By taking the first derivative of the polynomial in Theorem 14 at y = 1, we obtain the first redefined Zagreb index of the chain of silicates SC p q as follows: Let p be odd and p < q. Then the first redefined Zagreb index of SC p q is 5 2 pq + 7 6 p + 1 3 q + 1 6 .
Proof. Using the atom-bond partition from Table 3, in the formula of the second redefined Zagreb polynomial (7), we obtain

Conclusions
In the analysis of quantitative structure-property relationships (QSPRs) and (QSARs), chemical indices are major implements used to approximate the characteristic features of biological activities, and physical, biomedicine, and molecular compounds. It is ordinary for questions to emerge about the characterization of silicate networks on the bases of the nature of Zagreb polynomials. We computed Zagreb polynomials for the chain of silicates under various situations in this research article. We obtained the first Zagreb, second Zagreb, hyper-Zagreb, augmented Zagreb, redefined first Zagreb, redefined second Zagreb, and redefined third Zagreb indices for the chain of silicates SC p q from these Zagreb polynomials. For instance, topological indices or Zagreb indices are used to create quantitative structureactivity relationships (QSARs) that connect the chemical structure of molecules to the biological activities or other characteristics of such compounds. Open problems: For the characterization of the chain of silicates, followers are invited to discuss or research the following open problem: • Are Zagreb polynomials and Zagreb indices affected when both p and q are even or odd? • The results will be interesting when p ≥ q. Informed Consent Statement: Not applicable.

Data Availability Statement:
No data were used to support this study.

Conflicts of Interest:
The authors declare no conflict of interest.
Sample Availability: Not available.