Characterization of Temperature Indices of Silicates

A topological index is a numerical value that describes the entire structure of a chemical compound’s molecular graph and aids in the understanding of its physical properties, chemical reactivities, and boiling activities. In chemical graph theory, these indices are instrumental in quantifying various chemical properties of chemical compounds such as SiO4. The selection of SiO4 in this work is an important aspect because of its structural flexibility, easy availability, and extraordinary nature to find its numerical values. In this paper, we compute systematically the first and second temperature, hyper temperature indices, the sum connectivity temperature index, the product connectivity temperature index, the reciprocal product connectivity temperature index, and the F temperature index of a molecular graph SiO4 embedded in a chain. All of the discussed interlinked networks in this manuscript are well motivated by the molecular structure of SiO4.

physical, chemical, and thermal properties, biological activity, and chemical activity [9].Topological indices, which are molecular descriptors, can characterize these features and specific graphs [23].In chemical graph theory, vertices represent atoms in a chemical compound whereas edges indicate chemical bonding between the atoms [25].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 structure of chemical molecule [6,7,12].
Mathematical chemistry explains the way to use polynomials and functions to provide instructions hidden in the symmetry of molecular graphs, and graph theory has many applications in modern chemistry, particularly organic chemistry.Many applications of topological indices are used in theoretical chemistry, particularly qualitative structure property relationships (QSPR)/ qualitative structure activity relationships (QSAR) research.Many well-known researchers have investigated topological indices in order to learn more about various graph families [5,19].For example, in QSPR and QSAR, topological indices are employed as handy numerical descriptors in comparison with biological, physical and chemical parameters of molecules, which is an advantage for chemical industry.Many researchers have worked on various chemical compounds and computed topological descriptors of various molecular graphs during recent years [1,13,14,26].
Graph indices have found a number of applications in a wide range of fields in chemistry including chemical documentation, structure-property relationships, structureactivity relationships, and pharmaceutical drug design.The general calculation of graph indices has attracted a lot of interest in mathematics [17].
We only consider finite, simple, connected graphs in this paper.Assume that G is a graph with the vertex set V G and the edge set E G .The number of vertices adjacent to a vertex u determines its degree d u .For fundamental notations and terminologies, we refer the reader to [20].
Fajtlowicz put forward the follow definition of the temperature of any vertex u of a graph G [8]: The first temperature index [21] is introduced as follows: In 2020, Kulli introduced second temperature index [15], which is given by Kulli investigated the first and second hyper temperature indices in [15], which are defined as In the same paper [15], some other related topological indices are introduced.The sum connectivity temperature index, the product connectivity temperature index, and the reciprocal product connectivity index are defined respectively as Kulli also studied the F-temperature index and general temperature index of a graph G in [15], and they are defined as In industrial chemistry, a silicate Si is an element of a family of anions (an ion is a atom or molecule with a net-electrical charge) containing of silicon and oxygen, Boyer used the general formula SiO for 0 ≤ t < 2, in [4].Some researchers also explain the family of anions by using this formula, the Orthosilicate family SiO 4− 4 (t = 0), see in [3], metasilicate SiO 2− 3 (t = 1), see in [24] and pyrosilicate Si 2 O 6− 7 (t = 1 2 , n = 2), see in [11].We can extend silicate Si to any anions containing silicon (atom-bonding with other than oxygen atoms), as hexafluorosilicate SiF 2− 3 , see in [10].Here, we discuss only chain of silicates, which is obtained by alternating sequence of tetrahedron SiO 4 [18,22].
In this article, the above defined eight temperature indices are constructed by the atom-bonds partition of chain of silicates SC p q , which are partitioned according to the degrees of its Si and oxygen atoms.We also investigate silicon tetrahedron SiO 4 in a compound structure and derive the precise formulas of certain essential degreebased temperate indices using the approach of atom-bonds partitioning of molecular structure of silicates.We obtain the the concept of temperature indices from [2,15].

Chain of Silicates SC p q
The basic unit of silicates is empirically represented by formula SiO 4 , possessing tetrahedron geometry [16].Almost all the silicates contain SiO 4 tetrahedron.From chemical point of view, a tetrahedron SiO 4 , as shown in Fig. 1, containing oxygen atoms at the four corners of tetrahedron, and the silicon atom is bonded with equally spaced atoms of oxygen.From resulting SiO 4 , a silicate tetrahedron, joins with other SiO 4 horizontally, a single chain of silicates is obtained.Similarly, when two molecules of SiO 4 join corner to corner, then each SiO 4 shares its oxygen atoms with the other SiO 4 molecule, as in Fig. 1.After completing this process of sharing, these two molecules of SiO 4 , can be joined with two other molecules.Now, we get a chain of silicates SC p q , where p and q are the numbers of silicate chains formed and total number of SiO 4 in one silicate chain, respectively.Here, in chain of silicates SC p q , pq number of tetrahedron SiO 4 is used; see Fig. 1.Here, in chain of silicates SC p q , we observed that there are three type of atom-bonds on the bases of valency of every atom of SC p q .Therefore, there are two types of atoms v i and v j , such that d v i = 3 and d v j = 6, where d v i and d v j means the valency of atoms ∀ v i , v j ∈ SC p q .According to the valencies (3 and 6) of atoms, there are three types of atom-bonds, which are (3 ∼ 3), (3 ∼ 6) and (6 ∼ 6) in SC p q .On the base of valency, Table 1 provides the partition shown as: The total number of atoms and atom-bonds SC p q : |V (SC p q )|=3p 2 −p and |E(SC p q )|=6pq +3p−q By using Eq. 1 and above partition of SC p q , three types of edges on the basis of the temperature of end vertices of an edge can be identified.They are described in Table 1.

Temperature Indices for Chain of Silicates SC
p q p = q Theorem 2.1 Let SC p q be a chain of silicates.Then the first temperature index is 6(3p+2)  3p 2 −p−3 + 9(3p 2 +3p−4)(3p 2 −p−4) (3p 2 −p−3)(3p 2 −p−6) + 12(3p 2 −6p+2) Proof Using the atomic bonds partition from Table 1 in the formula of the first temperature index (2), we obtain After simplification, we get Theorem 2.2 Let SC p q be a chain of silicates.Then the second temperature index is Proof Using the atomic bonds partition from Table 1 in the formula of the second temperature index (3), we obtain After simplification, we get Theorem 2.3 Let SC p q be a chain of silicates.Then the first hyper temperature index is 36(3p+2)  (3p Proof Using the atomic bonds partition from Table 1 in the formula of the first hyper temperature index (4), we obtain After simplification, we get Theorem 2. 4 Let SC p q be a chain of silicates.Then the second hyper temperature index is 81(3p+2)  (3p 2 −p−3) 4 + 324(3p 2 +3p−4) Proof Using the atomic bonds partition from Table 1 in the formula of the second temperature index (5), we obtain After simplification, we get Theorem 2.5 Let SC p q be a chain of silicates.Then the sum connectivity temperature index is .
Proof Using the atomic bonds partition from Table 1 in the formula of the sum connectivity temperature index (6), we obtain After simplification, we get Theorem 2.6 Let SC p q be a chain of silicates.Then the product connectivity temperature index is
Proof Using the atomic bonds partition from Table 1 in the formula of the product connectivity temperature index (7), we obtain After simplification, we get Theorem 2.7 Let SC p q be a chain of silicates.Then the reciprocal product temperature index is 18(3p+2)  (3p Proof Using the atomic bonds partition from Table 1 in the formula of the second temperature index (8), we obtain After simplification, we get be a chain of silicates.Then the F-temperature index is Proof Using the atomic bonds partition from Table 1 in the formula of the F-temperature index (9), we obtain After simplification, we get In this section, we perform a numerical and graphical comparison temperature indices for n = 2, 3, 4, ..., 12 of chain of silicates SC p q .As we increase the value of p or q, the value of temperatures indices T 1 , T 1 , T 2 , H T 1 , RP T and F T decreased gradually while the numerical values of the temperature indices H T 2 , ST and P T increased gradually.These changes are also represented in the Figs. 2 and 3 (Table 2).

Conclusion
In this paper, we compute the first temperature index, second temperature index, first hyper temperature index, second hyper temperature index, sum temperature index, product temperature, reciprocal product temperature index, and F-temperature index of silicate network and silicate chain network, which correlates well with entropy, acentric factor, enthalpy of vaporization, and standard enthalpy of vaporization.In QSPR/QSAR research, topological indices such as the Zagreb index, Randic index, and atombond connectivity index are utilized to predict chemical compound bioactivity.This study disclosed new avenues in the field of topological index for silicates networks.

Open Problems
It is natural to arise a question about the characterization of chain of silicates on the basis of the nature of temperature indices.For the characterization of the chain of silicates, the following open problems seem to be interesting.
• Is the temperature indices effected, when both p and q are even or odd, one is even other is odd, p < q? • What about the case p ≥ q?

Fig. 1
Fig. 1 Schematic illustration of chain of silicates: where circular blue and black elliptical shapes showing the oxygen the silicon atoms, respectively

Fig. 2
Fig. 2 2D-comparison of temperature indices of SC p q

) 2 . 2
Numerical and Graphical Comparison of Temperature Indices for Chain of Silicates SC p q

Table 2
Temperature indices of chain of silicates SC p