Computation of Zagreb Polynomial and Indices for Silicate Network and Silicate Chain Network

Ghani, Muhammad Usman; Inc, Mustafa; Sultan, Faisal; Cancan, Murat; Houwe, Alphonse

doi:https://doi.org/10.1155/2023/9722878

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Computational and Theoretical Characteristics of Chemical Graph Theory and Applications

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Volume 2023 | Article ID 9722878 | https://doi.org/10.1155/2023/9722878

Computation of Zagreb Polynomial and Indices for Silicate Network and Silicate Chain Network

Muhammad Usman Ghani,¹Mustafa Inc,^2,3Faisal Sultan,¹Murat Cancan,⁴and Alphonse Houwe⁵

Academic Editor: Hassan Raza

Received25 Jun 2022

Accepted20 Aug 2022

Published23 May 2023

Abstract

The connection of Zagreb polynomials and Zagreb indices to chemical graph theory is a bifurcation of mathematical chemistry, which has had a crucial influence on the development of chemical sciences. Nowadays, the study of topological indices has become a vast effective research area in chemical graph theory. In this article, we add up eight different Zagreb polynomials for the Silicate Network and Silicate Chain Network. From these Zagreb polynomials, we catch up on degree-based Zagreb indices. We also provide a graphical representation of the outcome that describes the dependence of topological indices on the given parameters of polynomial structure.

1. Introduction

Using chemical graph theory, we can determine a wide range of characteristics such as chemical networks; physical, chemical, and thermal properties; biological activity; and chemical activity. Topological indices, which are molecular descriptors, can characterise these features and specific graphs [1, 2]. In chemical graph theory, vertices represent atoms and edges represent chemical bonding between the atoms [3, 4]. 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 [5–7].

Hayat et al. [8] and Ghani et al. [9] presented valency-based molecular descriptors for p-electronic measurements of lower polycyclic aromatic hydrocarbon energy. As a result of the incomplete combustion of organic matter, cyclic aromatic hydrocarbons () are widely diffused and relocated in the ecosystem. Many and their hydroxyl are very poisonous, toxic, and/or carcinogenic to bacteria as well as higher systems such as humans. In Qualitative Structure Property Relationships (QSPR) and Qualitative Structure Activity Relationships (QSAR), topological indices are used directly as simple numerical descriptors in comparison with physical, biological, or chemical parameters of molecules, which is an advantage in the chemical industry. Many researchers have worked on various chemical compounds and computed topological descriptors of various molecular graphs over the last few decades [10, 11].

In chemical graph theory, a molecular graph is a simple connected graph composed of chemical atoms and bonds, which are commonly referred to as vertices and edges, respectively, and there must be linkage between the vertices set and the edges set . The valency of each atom is actually the total number of atoms linked to of , and it is denoted by [12].

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 [13]. In 2004, Gutman and Das adulate characteristics of the first and second Zagreb polynomials for some chemical graphs of a chemical compound, which we studied in the research articles [14, 15]. The first Zagreb polynomial corresponding to the first Zagreb index is defined as

The second Zagreb polynomial, which corresponds to the second Zagreb index [14], is written as

In 2013, Shirdel et al. initiated the concept of the hyper Zagreb index [16]. The hyper Zagreb polynomial and index are defined as

The modified Zagreb polynomial and index [17] are defined as

In 2010, Furtula et al. introduced the augmented Zagreb index [18]. The augmented Zagreb polynomial and index are defined as

Ranjini, Lokesha, and Usha introduced a redefined version of the Zagreb indices , , and in 2013 [19]. The redefined Zagreb polynomial and indices are defined as follows:

A silicate is an element of a family of anions (an ion is an atom or molecule with a net-electrical charge) containing silicon and oxygen in industrial chemistry, usually represented by the general formula , where . Using this formula, the Orthosilicate family, see in [20], Metasilicate, see in [21] and Pyrosilicate, see in [22]. We can extend silicate to any anions containing silicon (atom-bonding with other than ), as Hexafluorosilicate , see in [23]. Here, we discuss only chain of silicates, which is obtained by alternating sequence of tetrahedron , see for details [24, 25].

In this article, the above defined eight Zagreb polynomials and Zagreb indices are constructed by the atom-bonds set of Silicate Network and Silicate Chain Network , which partitioned according to the valencies of its and atoms. We also investigate silicon tetrahedron in a compound structure and derived the precise formulas of certain essential valency-based Zagreb indices using the approach of atom-bonds partitioning of molecular structure of silicates.

2. Zagreb Polynomials and Indices of Silicate Network

Metal oxides or metal carbonates are fused with sand to form silicate networks. The basic unit of silicates is the tetrahedron , found in almost all silicates. The sides of the tetrahedron represent oxygen atoms, while the middle represents silicon atoms from a chemical perspective [26]. Figure 1 depicts a tetrahedron in a silicate network , where is the number of hexagons between the centre and the network’s boundary. A silicate sheet network is a collection of linked to other rings in a two-dimensional plane by shared oxygen atoms, resulting in a sheet-like structure, as shown in Figure 1.

Silicon atoms and corner atoms (lying on tetrahedrons in each ring) have valency 3 in the Silicate Network , whereas all other atoms have valency 6. The numbers of atoms of valency 3 and valency 6 are and , respectively. Thus, the total number of atoms and the total number of atom-bonds are shown in equation (9).

According to the valencies of the atoms, there are three types of atom-bonds in : (3,3), (3,6) and (6,6). The atom-bonds partition of is shown in Table 1.

Theorem 1. Let be a Silicate Network, then the first Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of the first Zagreb polynomial (1), we obtain This gives

By taking the first derivative of the polynomial in Theorem 17 at , we get the first Zagreb index of Silicate Network as follows:

Corollary 2. Let be a Silicate Network, then the first Zagreb index of is .

Theorem 3. Let be a Silicate Network, then the second Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of second Zagreb polynomial (2), we get This gives

By taking the first derivative of the polynomial in Theorem 19 at , we get the second Zagreb index of Silicate Network as follows:

Corollary 4. Let be a Silicate Network, then the second Zagreb index of is .

Theorem 5. Let be a Silicate Network, then the hyper Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of hyper Zagreb polynomial (3), we get This gives

By taking the first derivative of the polynomial in Theorem 21 at , we get the hyper Zagreb index of Silicate Network as follows:

Corollary 6. Let be a Silicate Network, then the hyper Zagreb index of is .

Theorem 7. Let be a Silicate Network, then the modified Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of modified Zagreb polynomial (4), we get This gives

By taking the first derivative of the polynomial in Theorem 23 at , we get the modified Zagreb index of Silicate Network as follows:

Corollary 8. Let be a Silicate Network, then the modified Zagreb index of is .

Theorem 9. Let be a Silicate Network, then the augmented Zagreb polynomial of is

Proof. Using the atom-bonds partition from Table 1 in the formula of augmented Zagreb polynomial (5), we get This gives

By taking the first derivative of the polynomial in Theorem 25 at , we get the augmented Zagreb index of Silicate Network as follows:

Corollary 10. Let be a Silicate Network, then the augmented Zagreb index of is .

Theorem 11. Let be a Silicate Network, then the first redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of first redefined Zagreb polynomial (6), we get This gives

By taking the first derivative of the polynomial in Theorem 27 at , we get the first redefined Zagreb index of Silicate Network as follows:

Corollary 12. Let be a Silicate Network, then the first redefined Zagreb index of is .

Theorem 13. Let be a Silicate Network, then the second redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of second redefined Zagreb polynomial (7), we get This gives

By taking the first derivative of the polynomial in Theorem 29 at , we get the second redefined Zagreb index of Silicate Network as follows:

Corollary 14. Let be a Silicate Network, then the second redefined Zagreb index of is .

Theorem 15. Let be a Silicate Network, then the third redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 1 in the formula of third redefined Zagreb polynomial (8), we get This gives

By taking the first derivative of the polynomial in Theorem 31 at , we get the third redefined Zagreb index of Silicate Network as follows:

Corollary 16. Let be a Silicate Network, then the third redefined Zagreb index of is .

3. Comparison

In this section, we present in Table 2 and Figure 2 a numerical and graphical comparison of the Zagreb indices of Zagreb polynomials for for the Silicate Network .

In the Silicate Network, the formulae for the polynomials , , , , , , , and show that the degree of the polynomial increases while the coefficients remain constant. As a result, the growing behaviour in each graph of their indices is consistent, although the expansion varies as the degree of the polynomial increases.

4. Zagreb Polynomials and Indices of Silicate Chain Network

In this section, we will look at a family of silicate chain networks denoted by and obtained by linearly arranging tetrahedral , as shown in Figure 3.

It can be seen in Silicate Chain Network , see Figure 3, that silicon atoms and corner atoms (lying on tetrahedrons in each ring) have valency 3, whereas all other atoms have valency 6 [27]. The number of atoms of valency 3 and valency 6 is and , respectively. Thus, the total number of atoms and total number of atom-bonds are shown in equation (26).

According to the valencies of the atoms, there are also three types of atom-bonds in : (3,3), (3,6), and (6,6). The atom-bonds partition of is shown in Table 3.

Theorem 17. Let be a Silicate Chain Network, then the first Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of first Zagreb polynomial (1), we get This gives

By taking the first derivative of the polynomial in Theorem 17 at , we get the first Zagreb index of Silicate Chain Network as follows:

Corollary 18. Let be a Silicate Chain Network, then the first Zagreb index of is .

Theorem 19. Let be a Silicate Chain Network, then the second Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of second Zagreb polynomial (2), we get This gives

By taking the first derivative of the polynomial in Theorem 19 at , we get the second Zagreb index of Silicate Chain Network as follows:

Corollary 20. Let be a Silicate Chain Network, then the second Zagreb index of is .

Theorem 21. Let be a Silicate Chain Network, then the hyper Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of hyper Zagreb polynomial (3), we get This gives

By taking the first derivative of the polynomial in Theorem 21 at , we get the hyper Zagreb index of Silicate Chain Network as follows:

Corollary 22. Let be a Silicate Chain Network, then the hyper Zagreb index of is .

Theorem 23. Let be a Silicate Chain Network, then the modified Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of modified Zagreb polynomial (4), we get This gives

By taking the first derivative of the polynomial in Theorem 23 at , we get the modified Zagreb index of Silicate Chain Network as follows:

Corollary 24. Let be a Silicate Chain Network, then the modified Zagreb index of is .

Theorem 25. Let be a Silicate Chain Network, then the augmented Zagreb polynomial of is

Proof. Using the atom-bonds partition from Table 3 in the formula of augmented Zagreb polynomial (5), we get This gives

By taking the first derivative of the polynomial in Theorem 25 at , we get the augmented Zagreb index of Silicate Chain Network as follows:

Corollary 26. Let be a Silicate Chain Network, then the augmented Zagreb index of is .

Theorem 27. Let be a Silicate Chain Network, then the first redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of first redefined Zagreb polynomial (6), we get This gives

By taking the first derivative of the polynomial in Theorem 27 at , we get the first redefined Zagreb index of Silicate Chain Network as follows:

Corollary 28. Let be a Silicate Chain Network, then the first redefined Zagreb index of is .

Theorem 29. Let be a Silicate Chain Network, then the second redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of second redefined Zagreb polynomial (7), we get This gives

By taking the first derivative of the polynomial in Theorem 29 at , we get the second redefined Zagreb index of Silicate Chain Network as follows:

Corollary 30. Let be a Silicate Chain Network, then the second redefined Zagreb index of is .

Theorem 31. Let be a Silicate Chain Network, then the third redefined Zagreb polynomial of is .

Proof. Using the atom-bonds partition from Table 3 in the formula of third redefined Zagreb polynomial (8), we get This gives

By taking the first derivative of the polynomial in Theorem 31 at , we get the third redefined Zagreb index of Silicate Chain Network as follows:

Corollary 32. Let be a Silicate Chain Network, then the third redefined Zagreb index of is .

5. Comparison

In this section, we present a numerical and graphical comparison of the Zagreb indices of Zagreb polynomials for , for the Silicate Chain Network in Table 4 and Figure 4.

These numerical variables correlate to a graph that is useful in linking the structure with various physicochemical attributes, chemical reactivity, and biological activities. The values of , , , , and in this graph (Figure 4) are rapidly increasing, while the values of , “,” and “” are slowly increasing.

6. Closing Remarks

In this article, two important silicon tetrahedron compound structures are considered, and the accurate formulas of some important valency-based topological indices are calculated using the technique of atom-bond partitioning of these molecular structures. Our investigated results, such as the Zagreb indies, are useful for determining physio-chemical properties of chemical compounds; as in 2005, Zhou explains in [28], such as formation enthalpies, boiling points, chromatographic retention times, vapour pressure, and surface areas. The obtained results are also innovative and noteworthy contributions to network science, providing a foundation for understanding the deep topology of these important networks. These findings may also be useful in determining the role of silicon-carbon in electronics and industry.

Data Availability

The data used to support the findings of this study are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Muhammad Usman Ghani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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