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\begin{document}
\title{Characterization of temperature indices of silicates}
\author{Abdul Rauf Khan$^{1},\,$ Muhammad Usman Ghani$^{2},\,$ Abdul Ghaffar$^{1},\,$  \\Hafiz Muhammad Asif$^{3},\,$ Mustafa Inc$^{4,5}$\\
\\
\\
$^{1}$\textit{Department of Mathematics, Faculty of Science, Ghazi University 32200, Dera Ghazi Khan, Punjab, Pakistan.}\\
$^{2}$\textit{Department of Mathematics, Khawaja Fareed University of Engineering, Information Technology, 64200, Rahim Yar Khan, Punjab, Pakistan}\\
$^{3}$\textit{DInstitute of Chemical Sciences, Baha Uddin Zakariya University, Multan, Punjab, Pakistan.}\\
$^{4}$\textit{Firat University, Science Faculty, Department of Mathematics, 23119
Elazig, Turkey}\\
$^{5}$\textit{Department of Medical Research, China Medical University, 40402 Taichung, Taiwan}\\
\\
\\
\textit{e-mail: khankts@gmail.com, abdulghaffar.jaffar@gmail.com,\,musmanghani85a@gmail.com,}\\
\textit{hmasifc@bzu.edu.pk,\,minc@firat.edu.tr}}

\maketitle
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    %%%%%% Abstract %%%%%%%
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\begin{abstract}
\noindent
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 $SiO_{4}$. The selection of $SiO_{4}$ 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 $SiO_{4}$ embedded in a chain. All of the discussed interlinked networks in this manuscript are well motivated by the molecular structure of $SiO_{4}$.
\noindent
\textbf{Keywords:} Temperature indices, $SiO_{4}$ embedded in a chain.
%{\color{red}}
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\end{abstract}
\section{Introduction}
Using chemical graph theory, one can determine a wide range of characteristics such as chemical networks, physical, chemical, and thermal properties, biological activity, and chemical activity \cite{gutman2021some}. Topological indices, which are molecular descriptors, can characterize these features and specific graphs \cite{sigarreta2021mathematical}. In chemical graph theory, vertices represent atoms in a chemical compound whereas edges indicate chemical bonding between the atoms \cite{wei2020some}. 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 \cite{KCdas1,ding2021degree,irfan2021m}.\\
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 \cite{costa2021chemical,mondal2021qspr}. 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 \cite{al2021downhill,javaid2017m,javaid2017topological,zakharov2021topological}.\\
Graph indices have found a number of applications in a wide range of fields in chemistry including chemical documentation, structure-property relationships, structure-activity relationships, and pharmaceutical drug design. The general calculation of graph indices has attracted a lot of interest in mathematics \cite{liu2021graph}. \\
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 \cite{monsalve2021vertex}.\\
Fajtlowicz put forward the follow definition of the temperature of any vertex $u$ of a graph $G$ \cite{fajtlowicz1988conjectures}:
\begin{equation}\label{T}
  T_{u_i} = \frac{d_{u_i}}{|V_G|-d_{u_i}} \,\,\qquad where \,\,\qquad \forall~~u_{i}\in V_G
\end{equation}
The first temperature index \cite{narayankarharmonic} is introduced as follows:
\begin{equation}\label{T1}
  T_{1}(G) = \sum_{uv\in E(G)} \left(T_u +T_v\right)
\end{equation}
In 2020, Kulli introduced second temperature index \cite{kullib}, which is given by
\begin{equation}\label{T2}
  T_{2}(G) = \sum_{uv\in E(G)} \left(T_u \times T_v\right)
\end{equation}
Kulli investigated the first and second hyper temperature indices in \cite{kullib}, which are defined as
\begin{equation}\label{HT1}
 HT_{1}(G) = \sum_{uv\in E(G)} \left(T_u +T_v\right)^{2}
\end{equation}
\begin{equation}\label{HT2}
  HT_{2}(G) = \sum_{uv\in E(G)} \left(T_u \times T_v\right)^{2}
\end{equation}
In the same paper \cite{kullib}, 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
\begin{equation}\label{ST}
 ST(G) = \sum_{uv\in E(G)}\frac{1}{ \sqrt{\left(T_u +T_v\right)}}
\end{equation}
\begin{equation}\label{PT}
 PT(G) = \sum_{uv\in E(G)}\frac{1}{ \sqrt{\left(T_{u} \times T_{v}\right)}}
\end{equation}
\begin{equation}\label{RPT}
 RPT(G) = \sum_{uv\in E(G)}{ \sqrt{\left(T_u \times T_v\right)}}
\end{equation}
Kulli also studied the F-temperature index and general temperature index of a graph $G$ in \cite{kullib}, and they are defined as\\
\begin{equation}\label{FT}
 FT(G) = \sum_{uv\in E(G)}\left(T_{u}^{2} +T_{v}^{2}\right)
\end{equation}
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 $\left[SiO{_{4-t}^{(4-2t)-}}\right]_{n}$ for $0\leq t<2$, in \cite{boyer1997synthesis}. Some researchers also explain the family of anions by using this formula, the Orthosilicate family $SiO_{4}^{4-}(t=0)$, see in \cite{banjare2021studies}, metasilicate $SiO_{3}^{2-}(t=1)$, see in \cite{soares2021new} and pyrosilicate $Si_{2}O_{7}^{6-}(t=\frac{1}{2}, n=2)$, see in \cite{hester2021quantum}. We can extend silicate $Si$ to any anions containing silicon (atom-bonding with other than oxygen atoms), as hexafluorosilicate $SiF_{3}^{2-}$, see in \cite{haddaji2021effect}. Here, we discuss only chain of silicates, which is obtained by alternating sequence of tetrahedron $SiO_{4}$ \cite{mandlimath2021synthesis,selvarani2021generalization}. \\
In this article, the above  defined eight temperature indices are constructed by the atom-bonds partition of chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$, which are partitioned according to the degrees of its $Si$ and oxygen atoms. We also investigate silicon tetrahedron $Si O_{4}$ in a compound structure and derive the precise formulas of certain essential degree-based temperate indices using the approach of atom-bonds partitioning of molecular structure of silicates. We obtain the the concept of temperature indices from \cite{balmaceda2021evaluation,kullib}.

\section{Chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$}
The basic unit of silicates is empirically represented by formula $SiO_{4}$, possessing tetrahedron geometry \cite{liebau2012structural}. Almost all the silicates contain $SiO_{4}$ tetrahedron. From chemical point of view, a tetrahedron $SiO_{4}$, as shown in Figure \ref{SC}, 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 Figure \ref{SC}. 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 $\mathcal{S}\mathcal{C}_{q}^{p}$, 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 $\mathcal{S}\mathcal{C}_{q}^{p}$, $pq$ number of tetrahedron $SiO_{4}$ is used; see Figure \ref{SC}.
\begin{figure}[H]
  \centering
\includegraphics[width=13cm]{cs.eps}
   \caption{Schematic illustration of chain of silicates: where circular blue and black elliptical shapes showing the oxygen the silicon atoms, respectively}\label{SC}
\end{figure}
Here, in chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$, we observed that there are three type of atom-bonds on the bases of valency of every atom of $\mathcal{S}\mathcal{C}_{q}^{p}$. 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 $\forall $ $v_i,v_{j}\in \mathcal{S}\mathcal{C}_{q}^{p}$. According to the valencies (3 and 6) of atoms, there are three types of atom-bonds, which are $(3\sim3)$, $(3\sim6)$ and $(6\sim6)$ in $\mathcal{S}\mathcal{C}_{q}^{p}$. On the base of valency, Table \ref{tab1} provides the partition of the set of atom-bonds. The atom-bonds partition of $\mathcal{S}\mathcal{C}_{q}^{p}$ shown as:
\begin{eqnarray*}
   %\nonumber % Remove numbering (before each equation)
E_{(3,3)} &=& \Big{\{}e=u\sim v,\forall ~u,v\in V(\mathcal{S}\mathcal{C}_{q}^{p} )\Big{|} d_{u}=3,d_{v}=3\Big{\}}, |E_{(3,3)}|=3p+2\\
     E_{(3,6)} &=& \Big{\{}e=u\sim v,\forall ~u,v\in V(\mathcal{S}\mathcal{C}_{q}^{p} )\Big{|} d_{u}=3,d_{v}=6\Big{\}},~|E_{(3,6)}|=3(pq+q)-4\\
      E_{(6,6)} &=& \Big{\{}e=u\sim v,\forall ~u,v\in V(\mathcal{S}\mathcal{C}_{q}^{p} )\Big{|} d_{u}=6,d_{v}=6\Big{\}},~|E_{(6,6)}|=3(pq-2q)+2.
  \end{eqnarray*}
  The total number of atoms and atom-bonds $\mathcal{S}\mathcal{C}_{q}^{p}$:
\begin{equation*}\label{0}
  |V(\mathcal{S}\mathcal{C}_q^p)| = 3p^2-p  \qquad\qquad and \qquad \qquad |E(\mathcal{S}\mathcal{C}_q^p)| = 6pq+3p-q
\end{equation*}
By using equation (\ref{T}) and above partition of $\mathcal{S}\mathcal{C}_{q}^{p}$, three types of edges on the basis of the temperature of end vertices of an edge can be identified. They are described in Table \ref{tab1}.
\begin{table}[H]
\center
\caption{Atomic-bond partition of $\mathcal{S}\mathcal{C}_{q}^{p}$, for $p = q$}\label{tab1}
\begin{tabular} {|c|c|c| c|}
 \hline
 $(T_u , T_v)$&$\left(\frac{3}{(3p^2-p)-3},\frac{3}{(3p^2-p)-3}\right)$  &$\left(\frac{3}{(3p^2-p)-3},\frac{6}{(3p^2-p)-6}\right)$  &$\left(\frac{6}{(3p^2-p)-6},\frac{6}{(3p^2-p)-6}\right)$   \\
 \hline
Frequency&  $3p+2$&$3(pq+q)-4$&$3(pq-2q)+2$\\
  \hline
\end{tabular}
\end{table}

\subsection{Temperature indices for chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$ $p = q$}


\begin{theorem}\label{thm1}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the first temperature index is $\frac{6(3p+2)}{3p^2-p-3}+\frac{9(3p^2+3p-4)(3p^2-p-4)}{(3p^2-p-3)(3p^2-p-6)}+\frac{12(3p^2-6p+2)}{3p^2-p-6}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the first temperature index (\ref{T1}), we obtain
\begin{eqnarray*}
T_{1}(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\left(T_3 +T_3 \right)+ \sum_{E_{(3, 6)}} \left(T_3 +T_6\right) + \sum_{E_{(6, 6)}} \left(T_6 +T_6\right)\\
&=&(3p+2)\left[\frac{3}{(3p^2-p)-3}+\frac{3}{(3p^2-p)-3} \right]\\
&+&(3p^2+3p-4)\left[\frac{3}{(3p^2-p)-3}+\frac{6}{(3p^2-p)-6} \right]\\
&+&(3p^2-6p+2)\left[\frac{6}{(3p^2-p)-6}+\frac{6}{(3p^2-p)-6} \right]
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{t1}
T_1(\mathcal{S}\mathcal{C}_{q}^{p})=&\frac{6(3p+2)}{3p^2-p-3}+\frac{9(3p^2+3p-4)(3p^2-p-4)}{(3p^2-p-3)(3p^2-p-6)}+\frac{12(3p^2-6p+2)}{3p^2-p-6}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm2}
Let $\mathcal{S}\mathcal{C}_{q}^{p}
$ be a chain of silicates. Then the second temperature index is $\frac{9(3p+2)}{(3p^2-p-3)^2}+\frac{18(3p^2+3p-4)(3p^2-p-4)}{(3p^2-p-3)(3p^2-p-6)}+\frac{36(3p^2-6p+2)}{(3p^2-p-6)^2}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the second temperature index (\ref{T2}), we obtain
\begin{eqnarray*}
T_{2}(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\left(T_3 \times T_3 \right)+ \sum_{E_{(3, 6)}} \left(T_3 \times T_6\right) + \sum_{E_{(6, 6)}} \left(T_6 \times T_6\right)\\
&=&(3p+2)\left[\frac{3}{(3p^2-p)-3}\times \frac{3}{(3p^2-p)-3} \right]\\
&+&(3p^2+3p-4)\left[\frac{3}{(3p^2-p)
-3}\times \frac{6}{(3p^2-p)-6} \right]\\
&+&(3p^2-6p+2)\left[\frac{6}{(3p^2-p)-6}\times \frac{6}{(3p^2-p)-6} \right]
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{t2}
T_2(\mathcal{S}\mathcal{C}_{q}^{p})=\frac{9(3p+2)}{(3p^2-p-3)^2}+\frac{18(3p^2+3p-4)}{(3p^2-p-3)(3p^2-p-6)}+\frac{36(3p^2-6p+2)}{(3p^2-p-6)^2}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm3}
Let $\mathcal{S}\mathcal{C}_{q}^{p}
$ be a chain of silicates. Then the first hyper temperature index is $\frac{36(3p+2)}{(3p^2-p-3)^2}+\frac{81(3p^2+3p-4)(3p^2-p-4)^2}{(3p^2-p-3)^2 (3p^2-p-6)^2}+\frac{144(3p^2-6p+2)}{(3p^2-p-6)^2}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the first hyper temperature index (\ref{HT1}), we obtain
\begin{eqnarray*}
HT_{1}(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\left(T_3 +T_3 \right)^{2}+ \sum_{E_{(3, 6)}} \left(T_3 +T_6\right)^{2} + \sum_{E_{(6, 6)}} \left(T_6 +T_6\right)^{2}\\
&=&(3p+2)\left[\frac{3}{(3p^2-p)-3}+\frac{3}{(3p^2-p)-3} \right]^{2}\\
&+&(3p^2+3p-4)\left[\frac{3}{(3p^2-p)
-3}+\frac{6}{(3p^2-p)-6} \right]^{2}\\
&+&(3p^2-6p+2)\left[\frac{6}{(3p^2-p)
-6}+\frac{6}{(3p^2-p)-6} \right]^{2}
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{Ht1}
HT_1(\mathcal{S}\mathcal{C}_{q}^{p})=\frac{36(3p+2)}{(3p^2-p-3)^2}+\frac{81(3p^2+3p-4)(3p^2-p-4)^2}{(3p^2-p-3)^2 (3p^2-p-6)^2}+\frac{144(3p^2-6p+2)}{(3p^2-p-6)^2}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm4}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the second hyper temperature index is $\frac{81(3p+2)}{(3p^2-p-3)^4}+\frac{324(3p^2+3p-4)}{(3p^2-p-3)^2 (3p^2-p-6)^2}+\frac{1296(3p^2-6p+2)}{(3p^2-p-6)^4}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the second temperature index (\ref{HT2}), we obtain
\begin{eqnarray*}
HT_{2}(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\left(T_3 \times T_3 \right)^{2}+ \sum_{E_{(3, 6)}} \left(T_3 \times T_6\right)^{2} + \sum_{E_{(6, 6)}} \left(T_6 \times T_6\right)^{2}\\
&=&(3p+2)\left[\frac{3}{(3p^2-p)-3}\times \frac{3}{(3p^2-p)-3} \right]^{2}\\
&+&(3p^2+3p-4)\left[\frac{3}{(3p^2-p)-3}\times \frac{6}{(3p^2-p)-6} \right]^{2}\\
&+&(3p^2-6p+2)\left[\frac{6}{(3p^2-p)
-6}\times \frac{6}{(3p^2-p)-6} \right]^{2}
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{Ht2}
HT_2(\mathcal{S}\mathcal{C}_{q}^{p}
)=\frac{81(3p+2)}{(3p^2-p-3)^4}+\frac{324(3p^2+3p-4)}{(3p^2-p-3)^2 (3p^2-p-6)^2}+\frac{1296(3p^2-6p+2)}{(3p^2-p-6)^4}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm5}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the sum connectivity temperature index is $\frac{(3p+2)\sqrt{(3p^2-p-3)}}{\sqrt{6}}+\frac{(3p^2+3p-4)\sqrt{(3p^2-p-3) (3p^2-p-6)}}{3\sqrt{(3p^2-p-4)}}+\frac{\sqrt{(3p^2-p-6)}}{2(3p^2-6p+2)\sqrt{3}}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the sum connectivity temperature index (\ref{ST}), we obtain
\begin{eqnarray*}
ST(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\frac{1}{\sqrt{\left(T_3 + T_3 \right)}}+ \sum_{E_{(3, 6)}} \frac{1}{\sqrt{\left(T_3 + T_6\right)}} + \sum_{E_{(6, 6)}} \frac{1}{\sqrt{\left(T_6 + T_6\right)}}\\
&=&(3p+2)\frac{1}{\sqrt{\left[\frac{3}{(3p^2-p)
-3}+ \frac{3}{(3p^2-p)-3} \right]}}\\
&+&(3p^2+3p-4)\frac{1}{\sqrt{\left[\frac{3}{(3p^2-p)-3}+ \frac{6}{(3p^2-p)-6} \right]}}\\
&+&(3p^2-6p+2)
\frac{1}{\sqrt{\left[\frac{6}{(3p^2-p)-6}+ \frac{6}{(3p^2-p)-6} \right]}}
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{St}
ST(\mathcal{S}\mathcal{C}_{q}^{p}
)=&\frac{(3p+2)\sqrt{(3p^2-p-3)}}{\sqrt{6}}+\frac{(3p^2+3p-4)\sqrt{(3p^2-p-3) (3p^2-p-6)}}{3\sqrt{(3p^2-p-4)}}\nonumber\\
&+\frac{\sqrt{(3p^2-p-6)}}{2(3p^2-6p+2)\sqrt{3}}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm6}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the product connectivity temperature index is $\frac{9p^3+3p^2-11p-6}{3}+\frac{(3p^2+3p-4)\sqrt{(3p^2-p-3)(3p^2-p-6)}}{3\sqrt{2}}+\frac{(3p^2-6p+2)(3p^2-p-6)}{6}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the product connectivity temperature index (\ref{PT}), we obtain
\begin{eqnarray*}
PT(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\frac{1}{\sqrt{\left(T_3 \times T_3 \right)}}+ \sum_{E_{(3, 6)}} \frac{1}{\sqrt{\left(T_3 \times T_6\right)}} + \sum_{E_{(6, 6)}} \frac{1}{\sqrt{\left(T_6 \times T_6\right)}}\\
&=&(3p+2)\frac{1}{\sqrt{\left[\frac{3}{(3p^2-p)-3}\times \frac{3}{(3p^2-p)-3} \right]}}\\
&+&(3p^2+3p-4)\frac{1}{\sqrt{\left[\frac{3}{(3p^2-p)-3}\times \frac{6}{(3p^2-p)-6} \right]}}\\
&+&(3p^2-6p+2)\frac{1}{\sqrt{\left[\frac{6}{(3p^2-p)-6}\times \frac{6}{(3p^2-p)
-6} \right]}}
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{Pt}\nonumber
PT(\mathcal{S}\mathcal{C}_{q}^{p})&=&\frac{9p^3+3p^2-11p-6}{3}+\frac{(3p^2+3p-4)\sqrt{(3p^2-p-3)(3p^2-p-6)}}{3\sqrt{2}}\\
&+&\frac{(3p^2-6p+2)(3p^2-p-6)}{6}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm7}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the reciprocal product temperature index is $\frac{18(3p+2)}{(3p^2-p-3)^2}+\frac{(3p^2+3p-4)(45p^4-60p^3-67p^2+48p+72)}{(3p^2-p-3)^2(3p^2-p-6)^2}+\frac{144(3p^2-6p+2)}{(3p^2-p-6)^2}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the second temperature index (\ref{RPT}), we obtain
\begin{eqnarray*}
RPT(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\sqrt{\left(T_3 \times T_3 \right)}+ \sum_{E_{(3, 6)}} \sqrt{\left(T_3 \times T_6\right)} + \sum_{E_{(6, 6)}} \sqrt{\left(T_6 \times T_6\right)}\\
&=&(3p+2)\sqrt{\left[\frac{3}{(3p^2-p)-3}\times \frac{3}{(3p^2-p)-3} \right]}\\
&+&(3p^2+3p-4)\sqrt{\left[\frac{3}{(3p^2-p)
-3}\times \frac{6}{(3p^2-p)-6} \right]}\\
&+&(3p^2-6p+2)\sqrt{\left[\frac{6}{(3p^2-p)-6}\times \frac{6}{(3p^2-p)-6} \right]}
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{t2}
RPT(\mathcal{S}\mathcal{C}_{q}^{p})=\frac{3(3p+2)}{(3p^2-p-3)}+\frac{3(3p^2+3p-4)\sqrt{2}}{\sqrt{(3p^2-p-3)(3p^2-p-6)}}+\frac{6(3p^2-6p+2)}{(3p^2-p-6)}.
\end{eqnarray}
\end{proof}
\begin{theorem}\label{thm8}
Let $\mathcal{S}\mathcal{C}_{q}^{p}$ be a chain of silicates. Then the F-temperature index is $\frac{18(3p+2)}{(3p^2-p-3)^2}+\frac{(3p^2+3p-4)(45p^4-60p^3-67p^2+48p+72)}{(3p^2-p-3)^2(3p^2-p-6)^2}+\frac{144(3p^2-6p+2)}{(3p^2-p-6)^2}$.
\end{theorem}
\begin{proof}
Using the atomic bonds partition from Table \ref{tab1} in the formula of the F-temperature index (\ref{FT}), we obtain
\begin{eqnarray*}
FT(\mathcal{S}\mathcal{C}_{q}^{p}) &=&  \sum_{E_{(3, 3)}}\left(T_{3}^{2} +T_{3}^{2} \right)+ \sum_{E_{(3, 6)}} \left(T_{3}^{2} +T_{6}^{2}\right) + \sum_{E_{(6, 6)}} \left(T_{6}^{2} +T_{6}^{2}\right)\\
&=&(3p+2)\left[\Big{\{}\frac{3}{(3p^2-p)-3}\Big{\}}^{2}+\Big{\{}\frac{3}{(3p^2-p)-3} \Big{\}}^{2}\right]\\
&+&(3p^2+3p-4)\left[\Big{\{}\frac{3}{(3p^2-p)-3}\Big{\}}^{2}+\Big{\{}\frac{6}{(3p^2-p)-6} \Big{\}}^{2}\right]\\
&+&(3p^2-6p+2)
\left[\Big{\{}\frac{6}{(3p^2-p)
-6}\Big{\}}^{2}+\Big{\{}\frac{6}{(3p^2-p)-6}\Big{\}}^{2} \right]
\end{eqnarray*}
After simplification, we get
\begin{eqnarray}\label{Ft}
FT(\mathcal{S}\mathcal{C}_{q}^{p})=&\frac{18(3p+2)}{(3p^2-p-3)^2}+\frac{(3p^2+3p-4)(45p^4-60p^3-67p^2+48p+72)}{(3p^2-p-3)^2(3p^2-p-6)^2}\nonumber\\
&+\frac{144(3p^2-6p+2)}{(3p^2-p-6)^2}.
\end{eqnarray}
\end{proof}

\subsection{Numerical and graphical comparison of temperature indices for chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$}
In this section, we perform a numerical and graphical comparison temperature indices for $n = 2,3,4,...,12$ of chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$.
\begin{table}[H]
\centering
\caption{Temperature indices of chain of silicates $\mathcal{S}\mathcal{C}_{q}^{p}$ for $p=q$}\label{tab3}
\begin{tabular} {|c| c| c| c| c| c| c| c| c|}
        \hline
         $p$&$T_1$ & $T_2$ & $HT_1$& $HT_2$& $ST$ &$PT$ &$ RPT$&$FT$ \\
    \hline
2&  36.86&  14.970& 75.95&  654.06& 19.01&  37.47&      17.65&  23.44\\
3&  22.05&  2.970&  13.04&  792.08& 67.064& 256.64&     12.22&  5.71\\
4&  19.9&   1.370&  5.88&   886.76& 153.16& 876.99&     10.73&  2.9\\
5&  17.45&  0.808&  3.439&  951.75& 187.92& 2208.36&    11.14&  1.81\\
6&  17.17&  0.538&  2.281&  999  &  481.76& 4647.35&    10.54&  1.25\\
7&  16.73&  0.387&  1.63&   1034.86&745.07& 8677.45&    10.42&  0.92\\
8&  16.44&  0.292&  1.23&   1063.01&1088.26&14869.11&   10.36&  0.71\\
9&  16.24&  0.228&  0.96&   1085.68&1521.71&23879.61&   10.32&  0.56\\
10& 16.08&  0.180&  0.77&   1104.34&2055.82&36453.22&   10.29&  0.46\\
11& 15.96&  0.150&  0.63&   1085.68&2700.99&53421.06&   10.27&  0.38\\
12& 15.86&  0.120&  0.53&   1104.34&3467.591&7501.22&   10.26&  0.32\\

        \hline
\end{tabular}
\end{table}
\begin{figure}[H]
  \centering
\includegraphics[width=12cm]{TTTT.eps}
   \caption{2D-comparison of temperature indices of $\mathcal{S}\mathcal{C}_{q}^{p}$}\label{2D}
\end{figure}
\begin{figure}[H]
  \centering
\includegraphics[width=12cm]{3DT.eps}
   \caption{3D-comparison of temperature indices of $\mathcal{S}\mathcal{C}_{q}^{p}$}\label{3D}
\end{figure}
\section{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 atom-bond connectivity index are utilized to predict chemical compound bioactivity. This study disclosed new avenues in the field of topological index for silicates networks.
\section{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.
\begin{itemize}
  \item Is the temperature indices effected, when both $p$ and $q$ are even or odd, one is even other is odd, $p<q$?
  \item What about the case $p\geq q$?
\end{itemize}
\noindent
{\bf Funding} Not Applicable.\\
{\bf Author Contributions} [Abdul Rauf Khan], [Muhammad Usman Ghani] and [Abdul Ghaffar] conceived the theoretical model. [Abdul Rauf Khan], [Hafiz Muhammad Asif] and [Mustafa Inc] performed the calculations and plotted the graphs. They analyzed and discussed the results with [Muhammad Usman Ghani] and [Hafiz Muhammad Asif].\\
{\bf Data Availability} All data generated or analysed during this study are included in this article.\\\\
{\bf Declarations}\\
{\bf Competing Interests} The authors declare no competing interests.\\
{\bf Ethics Approval} Not Applicable.\\
{\bf Consent to Participate} Not Applicable.\\
{\bf Consent for Publication} The authors declare that the figures and tables used in this paper are original and are not published anywhere.\\
{\bf Conflict of Interest} The authors declare that they have no conflict of interest


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\end{document}
