On statistical evaluation of reverse degree based topological indices for iron telluride networks

Abstract

In the context of graph theory and chemical graph theory, this research conducts a detailed mathematical investigation of reverse topological indices as they relate to iron telluride networks, clarifying their complex interactions. Graph theory is a branch of abstract mathematics that carefully studies the connections and structural features of graphs made up of edges and vertices. These theoretical ideas are expanded upon in chemical graph theory, which models molecular architectures with atoms acting as vertices and chemical bonds as edges. By extending these concepts, this work investigates the reverse topological indices in the context of Iron Telluride networks and outlines their significant effects on chemical reactivity, molecular topology and statistical modeling. By navigating intricate mathematical formalisms and algorithmic approaches, the analysis provides profound insights into the reactivity patterns and structural dynamics of Iron Telluride compounds, enhancing our knowledge of solid-state chemistry and materials science.

Introduction

Graph theory, a fundamental mathematical framework, provides a systematic approach to finding and evaluating the complicated interactions between multiple objects or components. Graph theory is an area of mathematics that investigates these structures, sometimes known as graphs¹. Chem-informatics is the synthesis of chemistry, technology, and graph theory. The corresponding molecular graph is used to relate the physio-chemical and structural properties of organic materials, along with a few useful graph invariants. Atoms and covalent bonds within a molecule are depicted as clusters of points and lines on a molecular graph².

A polynomial, a matrix, a series of numbers, or a numerical value can all be used to identify a graph. The molecular graph is a diagrammatic depiction of a chemical compound, where the atoms and the chemical link between them are represented as the nodes and edges, respectively³. A chemical network is converted into a number that characterizes the topology of the network, which serves as the foundation for the generation of topological indices. Topological indices of graphs can be classified into several main types, including degree-based, distance-based, and counting-related indices⁴. Let $G = (V, E)$ be a basic connected graph, where E is the graph’s edge set and V is its vertex set. The degree of a vertex $\varsigma$ is the number of edges that intersect with it and is denoted by $\Xi (\varsigma )$. Topological indices come in various types, the two most common being degree- and distance-based indices.⁵.

Within the field of mathematical analysis, we explore the deep nuances related to the topological indices of reverse degree in the context of the iron telluride network. Because of its observable structural characteristics and possible uses in superconductivity and thermoelectricity, iron telluride $({\rm FeTe}_2)$, is a hot topic in materials research and solid-state physics. In the context of $({\rm FeTe}_2)$, our work seeks to clarify these indices and their statistical characteristics. The reverse degree-based topological indices, which are numerical descriptors obtained from the $({\rm FeTe}_2)$ molecular graph and capture structural subtleties without explicit chemical considerations, are at the heart of our study. Our main goals are to characterize and evaluate these metrics relevant to the $({\rm FeTe}_2)$ network and perform statistical tests to clarify relationships between these metrics and measurable physical events related to $({\rm FeTe}_2)$⁶.

The graph theoretical depiction of chemical structures, in which atoms are portrayed as vertices and bonds as edges in a graph, is the source of these indices. Standard degree-based topological indices take into account the degree of every vertex in the molecular graph. The number of edges incident to a vertex indicates its degree, which indicates the degree of branching or connectedness at that particular atom⁷.

Conversely, the reciprocals of the vertices’ degrees are employed in reverse degree-based indices. This indicates that vertices contribute more to the index value at lower degrees and less at higher degrees. Reverse degrees are used to highlight the significance of less linked atoms in the chemical structure. These less linked atoms are frequently essential in defining the physicochemical behavior, biological activity, and reactivity of molecules⁸. Reverse degree-based indices emphasize the importance of terminal atoms and peripheral structural motifs, which may have a disproportionate impact on molecular characteristics, by focusing on the reciprocal of the degrees⁹. The reciprocal of vertex degrees and maybe other graph theoretical properties are used in mathematical procedures to compute reverse degree-based indices¹⁰. The concept of reverse vertex degree $\Re (g)$ was introduced by¹¹. In 2016, Ediz, & Cancan computed the reverse Zagreb indices of cartesian product of graphs¹².

These indices offer numerical representations of structural variability, branching patterns, and molecular complexity. They have been used in combinatorial library design, virtual screening, and property prediction, among other molecular modeling and drug design domains. Reverse degree-based topological indices, in summary, provide a way to represent the structural properties of molecules from a graph perspective, highlighting the significance of peripheral motifs and less linked atoms. In QSAR investigations, they are useful instruments that facilitate the forecast and comprehension of molecular characteristics and functions^13,14.

Reversing the general Randi index by Milan Randi¹⁵ yields:

$$\begin{aligned} \Re {R_\rho }(G)=\sum _{\varsigma \varepsilon \in {E}(G)}{[\Re \Xi (\varsigma ) \times \Re \Xi (\varepsilon )]^\rho }; \,\, \, \, \, \rho = 1, \, -1, \, \frac{1}{2}, \, -\frac{1}{2}. \end{aligned}$$

(1)

The reverse-atom bond connectivity index by Estrada et al.^16,17is :

$$\begin{aligned} \Re {ABC}(G)=\sum _{\varsigma \varepsilon \in {E}(G)}\sqrt{\frac{\Re \Xi (\varsigma ) +\Re \Xi (\varepsilon )-2}{\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )}}. \end{aligned}$$

(2)

The reverse geometric arithmetic by Vukicevic et al.^18,19 index is defined as:

$$\begin{aligned} \Re {GA}(G)= \sum _{\varsigma \varepsilon \in {E}(G)}{\frac{2\sqrt{\Re \Xi (\varsigma ) \times \Re \Xi (\varepsilon )}}{\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )}}. \end{aligned}$$

(3)

The reverse first and second Zagreb indices by Gutman^20,21 are defined as:

$$\begin{aligned} & \Re {M}_1(G)=\sum _{\varsigma \varepsilon \in {E}(G)}(\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )). \end{aligned}$$

(4)

$$\begin{aligned} & \quad \Re {M}_2(G)= \sum _{\varsigma \varepsilon \in {E}(G)}(\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )). \end{aligned}$$

(5)

The reverse hyper Zagreb index by Shirdel et al.²² is:

$$\begin{aligned} \Re {HM}(G)=\sum _{\varsigma \varepsilon \in {E}(G)}{[\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )]^2}. \end{aligned}$$

(6)

The reverse forgotten index by Furtula and Gutman^23,24 is defined as:

$$\begin{aligned} \Re {F}(G)=\sum _{\varsigma \varepsilon \in {E}(G)}{[\Re \Xi (\varsigma )^2+\Re \Xi (\varepsilon )^2]}. \end{aligned}$$

(7)

The reverse first multiple and reverse second multiple Zagreb degree-based indices by Ghorbani and Azimi²⁵ are defined as:

$$\begin{aligned} & \Re {PM}_1(G)=\prod _{\varsigma \varepsilon \in {E}(G)}[\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )]. \end{aligned}$$

(8)

$$\begin{aligned} & \quad \Re {PM}_2(G)=\prod _{\varsigma \varepsilon \in {E}(G)}[\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )]. \end{aligned}$$

(9)

A bibliometric analysis (Fig. 1) expertly illustrates the global interest in the study of degree-based topological indicators. This mosaic of national research efforts enhances the discipline of graph theory by fostering a deeper grasp of topological indices and their wide variety of applications (https://www.scopus.com). In Fig. 2, we have presented the degree-based topological indices keywords bibliometric analysis in several ways. The study’s findings show how extensively degree-based topological indices are talked about (https://www.scopus.com).

Fig. 1

Bibliometric analysis: different countries’ degree-based topological indices  (https://www.scopus.com).

Fig. 2

Bibliometric analysis: topological indices based on degree   (https://www.scopus.com).

The iron telluride network’s structure

Because of its special qualities, iron telluride $({\rm FeTe}_2)$ has attracted a lot of interest in the field of materials research. It is a layered material with a typical formula of $MX_2$, where M is a transition metal and X is a chalcogen (sulphur, selenium, or tellurium). This family of materials is known as transition metal dichalcogenides (TMDCs). The layered crystal structure of ${\rm FeTe}_2$ is made up of two layers of tellurium atoms encased in one layer of iron atoms. Iron ditelluride can be synthesised using a number of techniques, such as hydrothermal synthesis, chemical vapour transfer, and chemical vapour deposition²⁶. Iron ditelluride has attracted a lot of attention in the world of materials research due to its unique electrical and magnetic characteristics. It is a type-II superconductor since it exhibits both superconductivity and magnetism at the same time. This property makes it an attractive material for many applications, including quantum computers, spintronics, and energy storage²⁷.

Figure 3 illustrates the computation of the Iron Telluride $({\rm FeTe}_2)$ formulae using a unit cell, whereas Figure 4 shows a more general construction. To get the topological indices of iron telluride $({\rm FeTe}_2)$, the edge partition will be considered as follows: Table 1 shows the edge partition of iron telluride $({\rm FeTe}_2)$ when (m, n) is larger than or equal to 1. Using the formula

$$\begin{aligned} \Re \Xi (\varsigma ) = \Delta (G) - \Xi (\varsigma ) + 1. \end{aligned}$$

, where $\Delta (G)$ denotes the largest degree of a vertex in a graph, one may compute reverse degree based edge partition (see Table 2). Depending on the degree of each edge and vertex, the edge set, let’s say $E_1$, $E_2$, $E_3$, $E_4$, and $E_5$.

Fig. 3

The structure of iron telluride $({\rm FeTe}_2)$ for $n=m = 2$²⁷.

Fig. 4

The structure of iron telluride $({\rm FeTe}_2)$ for $m = 2$, $n=3$²⁷.

Results for iron telluride network $({\rm FeTe}_2)$

In the molecular graph of ${\rm FeTe}_2$ the number of vertices of degree 1 are $n+3$, degree 2 are $4m+n-3$, and degree 3 are $4mn-2m-n+1$. The order and size of $FeTe_{2}$ is $4mn+2m+n+1$ and $6mn+m$, respectively. Table 1 displays the edge partition of ${\rm FeTe}_2$. Table 2 displays the reverse degree-based edge partition of ${\rm FeTe}_2$.

Table 1 Edge partition of $FeTe_{2}$ established on degrees of terminal vertices.

Table 2 Reverse degree based edge partition of $FeTe_{2}$.

• Reverse general Randic index

By considering the Eq. (1) and Table 2, we make computation as below:

$$\begin{aligned} \Re {R_\rho }(FeTe_{2})=\sum _{\varsigma \varepsilon \in {E}(FeTe_{2})}{[\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )]^\rho }; \, \, \, \, \, \rho =1, -1, \frac{1}{2}, -\frac{1}{2}. \end{aligned}$$

${For} \rho =1x$;

$$\begin{aligned} \Re {R}_{1}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}{[\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )]} \\= & {(3 \times 2)}{(2)}+{(3 \times 1)}{(n+1)} \\+ & {(2 \times 2)}{((2(m-1)))}+{(2 \times 1)}{(2(-2+2m+n))} \\+ & {(1 \times 1)}{(3(2mn-n+1)-5m)}\\ \Re {R}_{1}(FeTe_{2})= & 6mn+11m+4n+2. \end{aligned}$$

${For} \rho =-1$;

$$\begin{aligned} \Re {R}_{-1}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}{\frac{1}{[\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )]}}\\= & \frac{2}{3\times 2}+\frac{n+1}{3\times 1}+\frac{(2(m-1))}{2\times 2}\\+ & \frac{2(-2+2m+n)}{2\times 1}+\frac{3(2mn-n+1)-5m}{1\times 1}\\ \Re {R}_{-1}(FeTe_{2})= & 6mn-2.5m-1.666667n+1.166667. \end{aligned}$$

${For} \rho =\frac{1}{2}$;

$$\begin{aligned} \Re {R}_\frac{1}{2}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}{\sqrt{\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )}}\\= & {(2)}\sqrt{3\times 2}+{(n+1)}\sqrt{3\times 1}+{((2(m-1)))}\sqrt{2\times 2}\\+ & {(2(-2+2m+n))}\sqrt{2\times 1}+{(3(2mn-n+1)-5m)}\sqrt{1\times 1}\\ \Re {R}_\frac{1}{2}(FeTe_{2})= & 6mn+4.656854m+1.560478n-0.025824. \end{aligned}$$

${{For}} \rho =-\frac{1}{2}$;

$$\begin{aligned} \Re {R}_{-\frac{1}{2}}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}{\frac{1}{\sqrt{\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )}}}\\= & \frac{2}{\sqrt{3\times 2}}+\frac{n+1}{\sqrt{3\times 1}}+\frac{(2(m-1))}{\sqrt{2\times 2}}\\+ & \frac{2(-2+2m+n)}{\sqrt{2\times 1}}+\frac{3(2mn-n+1)-5m}{\sqrt{1\times 1}}\\ \Re {R}_{-\frac{1}{2}}(FeTe_{2})= & 6mn-1.171573m-1.008436n+0.56542. \end{aligned}$$

The numerical values of the previous determined result are i shown in Table 3 together the graphical behaviour is presented in Fig. 5.

Table 3 Numerical pattern of $\Re {R}_1(FeTe_{2})$, $\Re {R}_{-1}(FeTe_{2})$, $\Re {R}_{\frac{1}{2}}(FeTe_{2})$, $\Re {R}_{\frac{-1}{2}}(FeTe_{2})$.

Fig. 5

Comparison graph of $\Re {R_{1}}(FeTe_{2})$, $\Re {R_{-1}}(FeTe_{2})$, $\Re {R_\frac{1}{2}}(FeTe_{2})$, $\Re {R_\frac{-1}{2}}(FeTe_{2})$  (https://www.originlab.com/).

• Reverse atom bond connectivity index

By considering Eq. (2) and Table 2, we make computation as below:

$$\begin{aligned} \Re {ABC}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}\sqrt{\frac{\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )-2}{\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )}}\\= & {(2)}{\sqrt{\frac{3+2-2}{3\times 2}}}+{(n+1)}{\sqrt{\frac{3+1-2}{3\times 1}}}{((2(m-1)))}+{\sqrt{\frac{2+2-2}{2\times 2}}}\\+ & {(2(-2+2m+n))}{\sqrt{\frac{2+1-2}{2\times 1}}}+{(3(2mn-n+1)-5m)}{\sqrt{\frac{1+1-2}{1\times 1}}}\\ \Re {ABC}(FeTe_{2})= & 4.242641m+2.23071n-2.011931. \end{aligned}$$

• Reverse geometric arithmetic index

By considering the Eq. (3) and Table 2, we make computation as below:

$$\begin{aligned} \Re {GA}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}{\frac{2\sqrt{\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )}}{\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )}}\\ \\= & {\frac{2\sqrt{3\times 2}}{3+2}}{(2)}+{\frac{2\sqrt{3\times 1}}{3+1}}{(n+1)}+{\frac{2\sqrt{2\times 2}}{2+2}}{((2(m-1)))}\\+ & {\frac{2\sqrt{2\times 1}}{2+1}}{(2(-2+2m+n))}+{\frac{2\sqrt{1\times 1}}{1+1}}{(3(2mn-n+1)-5m)}\\ \Re {GA}(FeTe_{2})= & 6mn+0.771236m-0.248357n+0.054381. \end{aligned}$$

• Reverse first Zagreb index

By considering the Eq. (4) and Table 2, we make computation as below:

$$\begin{aligned} \Re {M}_{1}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}(\Re \Xi (\varsigma )+\Re \Xi (\varepsilon ))\\= & {(2)}{(3+2)}+{(n+1)}{(3+1)}+{((2(m-1)))}{(2+2)}\\+ & {(2(-2+2m+n))}{(2+1)}+{(3(2mn-n+1)-5m)}{(1+1)}\\ \Re {M}_{1}(FeTe_{2})= & 12mn+10m+4n. \end{aligned}$$

• Reverse second Zagreb index

By considering the Eq. (5) and Table 2, we make computation as below:

$$\begin{aligned} \Re {M}_{2}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}{(\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon ))}\\= & {(2)}{(3\times 2)}+{(n+1)}{(3\times 1)}+{((2(m-1)))}{(2\times 2)}\\+ & {(2(-2+2m+n))}{(2\times 1)}+{(3(2mn-n+1)-5m)}{(1\times 1)}\\ \Re {M}_{2}(FeTe_{2})= & 6mn+11m+4n+2. \end{aligned}$$

The numerical values of the previous determined result are i shown in Table 4 together the graphical behaviour is presented in Fig. 6.

Table 4 Numerical pattern of $\Re {ABC}(FeTe_{2})$, $\Re {GA}(FeTe_{2})$, $\Re {M_1}(FeTe_{2})$, $\Re {M_2}(FeTe_{2})$.

Fig. 6

Comparison graph of $\Re {ABC}(FeTe_{2})$, $\Re {GA}(FeTe_{2})$, $\Re {M_1}(FeTe_{2})$, $\Re {M_2}(FeTe_{2})$  (https://www.originlab.com/).

• Reverse hyper Zagreb index

By considering the Eq. (6) and Table 2, we make computation as below:

$$\begin{aligned} \Re {HM}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}} (FeTe_{2})}{[\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )]^2}\\= & {(2)}{(3+2)^2}+{(n+1)}{(3+1)^2}+{((2(m-1)))}{(2+2)^2}\\+ & {(2(-2+2m+n))}{(2+1)^2}+{(3(2mn-n+1)-5m)}{(1+1)^2}\\ \Re {HM}(FeTe_{2})= & 24mn+48m+22n+10. \end{aligned}$$

• Reverse First Multiple Zagreb Index

By considering the Eq. (8) and Table 2, we make computation as below:

$$\begin{aligned} \Re {PM}_1(FeTe_{2})= & \prod \limits _{i=1}^{5}\prod \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}[\Re \Xi (\varsigma )+\Re \Xi (\varepsilon )] \\ = & {(2)}{(3+2)} \times {(n+1)}{(3+1)}\times {(2(-2+2m+n))}{(2+1)}\\\times & {((2(m-1)))}{(2+2)}\times {(3(2mn-n+1)-5m)}{(1+1)} \\ \Re {PM}_1(FeTe_{2})= & 46080m^3n^2+23040m^2n^3+7680m^3n-111360m^2n^2\\- & 34560mn^3-38400m^3-34560m^2n+88320mn^2+11520n^3\\+ & 99840m^2+38400mn-23040n^2-84480m-11520n+23040. \end{aligned}$$

• Reverse second multiple Zagreb index

By considering the Eq. (9) and Table 2, we make computation as below:

$$\begin{aligned} \Re {PM}_2(FeTe_{2})= & \prod \limits _{i=1}^{5}\prod \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}[\Re \Xi (\varsigma )\times \Re \Xi (\varepsilon )]\\= & {(2)}{(3\times 2)}\times {(n+1)}{(3\times 1)}{((2(m-1)))}{(2\times 2)}\\\times & \times {(2(-2+2m+n))}{(2\times 1)}\times {(3(2mn-n+1)-5m)}{(1\times 1)}\\ \Re {PM}_2(FeTe_{2})= & 13824m^3n^2+6912m^2n^3+2304m^3n-33408m^2n^2-10368mn^3\\- & 11520m^3-10368m^2n+26496mn^2+3456n^3+29952m^2\\+ & 11520mn-6912n^2-25344m-3456n+6912. \end{aligned}$$

• Reverse forgotten index

By considering the Eq. (7) and Table 2, we make computation as below:

$$\begin{aligned} \Re {F}(FeTe_{2})= & \sum \limits _{i=1}^{5}\sum \limits _{\varsigma \varepsilon \in {E_{i}}(FeTe_{2})}{[\Re \Xi (\varsigma )^2+\Re \Xi (\varepsilon )^2]} \\= & {(2)}{(3^2+2^2)}+{(n+1)}{(3^2+1^2)}+{((2(m-1)))}{(2^2+2^2)}+\\+ & {(2(-2+2m+n))}{(2^2+1^2)}+{(3(2mn-n+1)-5m)}{(1^2+1^2)}\\ \Re {F}(FeTe_{2})= & 12mn+26m+14n+6. \end{aligned}$$

The numerical values of the previous determined result are i shown in Table 5 together the graphical behaviour is presented in Fig. 7.

Table 5 Numerical pattern of $\Re {HM}(FeTe_{2})$, $\Re {PM_1}(FeTe_{2})$, $\Re {PM_2}(FeTe_{2})$.

Fig. 7

Comparison graph of $\Re {HM}(FeTe_{2})$, $\Re {PM_1}(FeTe_{2})$, $\Re {PM_2}(FeTe_{2})$, $\Re {F}(FeTe_{2})$   (https://www.originlab.com/).

Rational curve fitting (${\mathfrak {R}}_{{c}{f}}$mn) for reverse topological indices

This section describes the methods for determining the links between the graphical properties of the linked chemical graph and the thermodynamic parameters of Iron Telluride $({\rm FeTe}_2)$. Then, the HoF of Iron Telluride $({\rm FeTe}_2)$ is calculated for various formula unit cells of Iron Telluride. The change in “enthalpy of formation,” or (HoF) is the transformation that occurs when a mole of a molecule is broken down into its component parts in their natural state²⁸. This transformation usually occurs at a specific temperature and pressure (usually $25C + 1$ atmosphere). The notation ‘$\delta H_f$’ is commonly used to refer to the HoF. The HoF is expressed in terms of the energy per mole (inkilojoules/mol) or kilocalories/mol(inkcal/mol) of ${\rm FeTe}_2$. Divide ${\rm FeTe}_2 = -51.9/ -65.8 kJ/mol$ by Avogadro’s number, $6.02214\times 10^{23}\, \textrm{mol}^{-1}$.

Topological indices are shown graphically for different formula unit cells. When fitting rational graphical models, the output variable is enthalpy, and the input variables are topological co-indices. Last but not least, many of the curves are fitted with the help of the curve fitting tool in MATLAB. Most of our built-in curve fitting techniques are used on our data. Let’s say we have a data collection and we have a number of observations of (n). Let’s also let’s say g is a set of all the fitted values that match Y. Let’s also think about the standard deviation. This is a key component that tells us how far our values differ from the mean. We can get a more precise fit with the standard deviation. To compute an error, let’s use the standard deviation. It can be expressed as a square root of the error value. Here, $\backslash RMSE$” stands for “standard deviation of residuals.” This test tells us how far the residuals differ from the model’s predicted values. It tells us how far away the residuals are from the mean. It’s easy to read this test as a mean squared error because it’s just Total squared error An extra statistical test is (SSE). The degree to which observed values differ from our fitted curve is examined using the $R^2$-test. A good match is shown by $R^2$ nearing 1, whereas a poor estimate is indicated by $R^2$ approaching 0. The ratio of estimated variance to actual variance is denoted by $R^2$. We will only investigate these three statistical tests, despite the fact that there are a few more in the literature, because MATLAB’s Rational Curve Fitting tool selects the model based on and only²⁹.

Below are the models that we have discussed for indices and correlation HOF. We have also used MATLAB to display the graphic representations as well as the curve that fits the statistical parameters. Figures 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19. We constructed rational curves using MATLAB’s curve fitting toolkit.models using Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. MATLAB tools were used to produce key performance measures, such as $\mathbb {RMSE}$, $\mathbb {SSE}$, $\mathfrak {r^2}$, and $adj\mathfrak {(r^2)}$, to evaluate the explanatory power and accuracy of the fitted models.

• HOF using $\Re {R_1}(FeTe_{2})$

$$\begin{aligned} f(\Re {R_1}) =\frac{(\chi _1\times \Re {R_1}^3 + \chi _2\times \Re {R_1}^2 + \chi _3\times \Re {R_1} + \chi _4)}{(\Re {R_1}^3 + \Psi _1\times \Re {R_1}^2 + \Psi _2\times \Re {R_1} + \Psi _3)} \end{aligned}$$

In which the Normalized mean of $\Re {R_1}$ is 222.5 and the standard deviation is 171.6. The coefficients are: $\kappa _1= -2.431$, with ${\mathbb {C}}_{b}=(-4.717, -0.1442)$, $\Psi _1= 1.233$ with ${\mathbb {W}}_{b}=(0.5785, 1.889)$, $\Psi _2= 0.196$ with ${\mathbb {W}}_{b}=(-0.2597, 0.6518)$, $\chi _2= -2.356$, with ${\mathbb {W}}_{b}=(-6.567, 1.854)$, $\kappa _3= 0.2201$ with ${\mathbb {W}}_{b}=(-1.498, 1.938)$, $\kappa _4= 0.1682$,with ${\mathbb {W}}_{b}=(-1.803, 2.14)$, $\Psi _3= -0.026$ with ${\mathbb {W}}_{b}=(-0.2623, 0.2103)$.

Fig. 8

HOF of $\Re {R_{1}}(FeTe_{2})$.

Table 6 HOF and $\Re {R_{1}}(FeTe_{2})$.

• HOF using $\Re {R_{-1}(FeTe_{2})}$

$$\begin{aligned} f(\Re {R_{-1}}) =\frac{(\kappa _1\times \Re {R_{-1}}^3 + \kappa _2\times \Re {R_{-1}}^2 + \kappa _3\times \Re {R_{-1}} + \kappa _4)}{(\Re {R_{-1}}^3 + \Psi _1\times \Re {R_{-1}}^2 + \Psi _2\times \Re {R_{-1}} + \Psi _3)} \end{aligned}$$

In which Normalized mean of $\Re {R_{-1}}$ is 135.4 and standard deviation is 125.6. The coefficients are: $\kappa _1= -2.385$, with ${\mathbb {W}}_{b}=(-4.822, 0.05173)$, $\kappa _2= -2.581$, with ${\mathbb {W}}_{b}=(-6.816, 1.655)$, $\Psi _1= 1.33$ with ${\mathbb {W}}_{b}=(0.7634, 1.896)$, $\kappa _3= -0.1022$, with ${\mathbb {W}}_{b}=(-1.848, 1.643)$,$\kappa _4= 0.1776$, with ${\mathbb {W}}_{b}=(-1.616, 1.971)$, $\Psi _2= 0.34$ with ${\mathbb {W}}_{b}=(-0.07087, 0.7509)$, $\Psi _3= -0.008958$ with ${\mathbb {W}}_{b}=(-0.2082, 0.1903)$.

Table 7 CF between HOF and $\Re {R_{-1}}(FeTe_{2})$.

Fig. 9

HOF of $\Re {R_{-1}(FeTe_{2})}$.

• HOF using $\Re {R_\frac{1}{2}(FeTe_{2})}$

$$\begin{aligned} f(\Re {R_\frac{1}{2}}) =\frac{(\kappa _1\times \Re {R_\frac{1}{2}}^3 + \kappa _2\times \Re {R_\frac{1}{2}}^2 + \kappa _3\times \Re {R_\frac{1}{2}} + \kappa _4)}{(\Re {R_\frac{1}{2}}^3 + \Psi _1\times \Re {R_\frac{1}{2}}^2 + \Psi _2\times \Re {R_\frac{1}{2}} + \Psi _3)} \end{aligned}$$

In which normalized mean of $\Re {R_\frac{1}{2}}$ is 181 and standard deviation is 150.4. The coefficients are: $\kappa _1= -2.413$, with ${\mathbb {W}}_{b}=(-4.752, -0.07324)$, $\kappa _2= -2.442$, with $\Psi _1= 1.27$ with ${\mathbb {W}}_{b}=(0.6477, 1.893)$, $\Psi _2= 0.2513$ with ${\mathbb {W}}_{b}=(-0.1896, 0.6922)$, ${\mathbb {W}}_{b}=(-6.663, 1.779)$, $\kappa _3= 0.09659$ with ${\mathbb {W}}_{b}=(-1.63, 1.823)$, $\kappa _4= 0.1742$ with ${\mathbb {W}}_{b}=(-1.728, 2.076)$, $\Psi _3= -0.0204$ with ${\mathbb {W}}_{b}=(-0.2414, 0.2006)$.

Fig. 10

HOF of $\Re {R_\frac{1}{2}(FeTe_{2})}$.

Table 8 CF between HOF and $\Re {R_\frac{1}{2}(FeTe_{2})}$.

• HOF using $\Re {R_\frac{-1}{2}(FeTe_{2})}$

$$\begin{aligned} f(\Re {R_\frac{-1}{2}}) = \frac{(\kappa _1\times \Re {R_\frac{-1}{2}}^3 + \kappa _2\times \Re {R_\frac{-1}{2}}^2 + \kappa _3\times \Re {R_\frac{-1}{2}} + \kappa _4)}{(\Re {R_\frac{-1}{2}}^3 + \Psi _1\times \Re {R_\frac{-1}{2}}^2 + \Psi _2\times \Re {R_\frac{-1}{2}} + \Psi _3)} \end{aligned}$$

Fig. 11

HOF of $\Re {R_\frac{-1}{2}}(FeTe_{2})$.

In which Normalized mean of $\Re {R_\frac{-1}{2}}$ is 143.8 and standard deviation is 130.3. The coefficients are: $\kappa _1= -2.391$, with ${\mathbb {W}}_{b}=(-4.805, 0.02294)$, $\kappa _2= -2.55$ with $\Psi _1= 1.317$ with ${\mathbb {W}}_{b}=(0.7374, 1.896)$, ${\mathbb {W}}_{b}=(-6.783, 1.683)$, $\kappa _3= -0.05842$, with ${\mathbb {W}}_{b}=(-1.799, 1.682)$, $\kappa _4= 0.1775$, with ${\mathbb {W}}_{b}=(-1.64, 1.995)$, $\Psi _2= 0.3205$ with ${\mathbb {W}}_{b}=(-0.09767, 0.7386)$, $\Psi _3= -0.01173$ with ${\mathbb {W}}_{b}=(-0.2154, 0.192)$.

Table 9 HOF and $\Re {R_\frac{-1}{2}}(FeTe_{2})$.

• HOF using $\Re {M_1}(FeTe_{2})$

Fig. 12

HOF of $\Re {M_1}(FeTe_{2})$.

$$\begin{aligned} f(\Re {M_1}) = \frac{(\kappa _1\times \Re {M_1}^3 + \kappa _2\times \Re {M_1}^2 + \kappa _3\times \Re {M_1} + \kappa _4)}{(\Re {M_1}^3 + \Psi _1\times \Re {M_1}^2 + \Psi _2\times \Re {M_1} + \Psi _3)} \end{aligned}$$

In which Normalized mean of $\Re {M_1}$ is 369 and standard deviation is 304.6. The coefficients are: $\kappa _1= -2.415$, with ${\mathbb {W}}_{b}=(-4.749, -0.08057)$, $\kappa _2= -2.434$, with ${\mathbb {W}}_{b}=(-6.653, 1.786)$, $\Psi _1= 1.267$ with ${\mathbb {W}}_{b}=(0.6407, 1.892)$, $\kappa _3= 0.109$ with ${\mathbb {W}}_{b}=(-1.617, 1.835)$, $\kappa _4= 0.1737$, with ${\mathbb {W}}_{b}=(-1.735, 2.083)$, $\Psi _2= 0.2458$ with ${\mathbb {W}}_{b}=(-0.1968, 0.6883)$, $\Psi _3= -0.02102$ with ${\mathbb {W}}_{b}=(-0.2435, 0.2014)$.

Table 10 HOF and $\Re {M_1}(FeTe_{2})$.

• HOF using $\Re {M_2}(FeTe_{2})$

$$\begin{aligned} f(\Re {M_2}) = \frac{(\kappa _1\times \Re {M_2}^3 + \kappa _2\times \Re {M_2}^2 + \kappa _3\times \Re {M_2} + \kappa _4)}{(\Re {M_2}^3 + \Psi _1\times \Re {M_2}^2 + \Psi _2\times \Re {M_2} + \Psi _3)} \end{aligned}$$

In which the Normalized mean of $\Re {M_2}$ is 222.5 and the standard deviation is 171.6. The coefficients are: $\kappa _1= -2.431$, with ${\mathbb {W}}_{b}=(-4.717, --0.1442)$, $\kappa _2= -2.356$, $\Psi _1= 1.233$ with ${\mathbb {W}}_{b}=(-6.567, 1.854)$, $\kappa _3= 0.2201$ with ${\mathbb {W}}_{b}=(-1.498, 1.938)$, $\kappa _4= 0.1682$ with ${\mathbb {W}}_{b}=(-1.803, 2.14)$, with ${\mathbb {W}}_{b}=(0.5785, 1.888)$, $\Psi _2= 0.196$ with ${\mathbb {W}}_{b}=(-0.2597, 0.6518)$, $\Psi _3= -0.026$ with ${\mathbb {W}}_{b}=(-0.2623, 0.2103)$.

Fig. 13

HOF of $\Re {M_2}(FeTe_{2})$.

Table 11 CF between HOF and $\Re {M_2}(FeTe_{2})$.

• HOF using $\Re {HM}(FeTe_{2})$

$$\begin{aligned} f(\Re {HM}) =\frac{(\kappa _1\times \Re {HM}^3 + \kappa _2\times \Re {HM}^2 + \kappa _3\times \Re {HM} + \kappa _4)}{(\Re {HM}^3 + \Psi _1\Re {HM}^2+\Psi _2\Re {HM}+\Psi _3)} \end{aligned}$$

In which the normalized mean of $\Re {HM}$ is 937 and the standard deviation is 710.4. The coefficients are: $\kappa _1= -2.435$, with ${\mathbb {W}}_{b}=(-4.709, -0.1606)$, $\Psi _1= 1.224$ with ${\mathbb {W}}_{b}=(0.562, 1.887)$, $\Psi _2= 0.1827$ with ${\mathbb {W}}_{b}=(-0.2763, 0.6416)$,$\kappa _2= -2.336$, with ${\mathbb {W}}_{b}=(-6.543, 1.872)$, $\kappa _3= 0.2499$, with ${\mathbb {W}}_{b}=(-1.466, 1.966)$, $\kappa _4= 0.1663$, with ${\mathbb {W}}_{b}=(-1.822, 2.155)$, $\Psi _3= -0.02718$ with ${\mathbb {W}}_{b}=(-0.2674, 0.2131)$.

Fig. 14

HOF of $\Re {HM}(FeTe_{2})$.

Table 12 CF between HOF and $\Re {HM}(FeTe_{2})$.

• HOF using $\Re {ABC}(FeTe_{2})$

$$\begin{aligned} f(\Re {ABC}) =\frac{(\kappa _1\times \Re {ABC}^5 + \kappa _2\times \Re {ABC}^4 + \kappa _3\times \Re {ABC}^3 + \kappa _4\times \Re {ABC}^2 + \kappa _5\times \Re {ABC} + \kappa _6)}{(\Re {ABC} + \Psi _1)} \end{aligned}$$

Fig. 15

HOF of $\Re {ABC}(FeTe_{2})$.

In which the normalized mean of $\Re {ABC}$ is 27.12 and the standard deviation is 15.86. The coefficients are: $\kappa _1= -2.051$, with ${\mathbb {W}}_{b}=(-2.489, -1.614)$, $\kappa _2= 1.785$ with ${\mathbb {W}}_{b}=(1.312, 2.258)$, $\kappa _5= -4.27$ with ${\mathbb {W}}_{b}=(-4.736, -3.805)$, $\kappa _6= 2.654$ with ${\mathbb {W}}_{b}=(2.254, 3.054)$, $\kappa _=3 4.811$ with ${\mathbb {W}}_{b}=(3.783, 5.839)$, $\kappa _4= -3.963$ with ${\mathbb {W}}_{b}=(-4.964, -2.962)$, $\Psi _1= -0.3968$, with ${\mathbb {W}}_{b}=(-0.4378, -0.3559)$.

Table 13 CF between HOF and $\Re {ABC}(FeTe_{2})$.

• HOF using $\Re {GA}(FeTe_{2})$

$$\begin{aligned} f(\Re {GA}) =\frac{(\kappa _1\times \Re {GA}^3 + \kappa _2\times \Re {GA}^2 + \kappa _3\times \Re {GA} + \kappa _4)}{(\Re {GA}^3 + \Psi _1\times \Re {GA}^2 + \Psi _2\times \Re {GA} + \Psi _3)} \end{aligned}$$

In which the normalized mean of $\Re {GA}$ is 155.4 and the standard deviation is 136.7. The coefficients are: $\kappa _1= -2.399$, with ${\mathbb {W}}_{b}= (-4.785, -0.01203)$, $\kappa _2= -2.512$, with ${\mathbb {W}}_{b}=(-6.741, 1.1717)$, $\Psi _1= 1.3$ with ${\mathbb {W}}_{b}=(0.7053, 1.895)$, $\Psi _2= 0.2961$, with ${\mathbb {W}}_{b}=(-0.1307, 0.7228)$,$\kappa _3= -0.00363$, with ${\mathbb {W}}_{b}=(-1.739, 1.731)$, $\kappa _4= 0.1768$, with ${\mathbb {W}}_{b}=(-1.67, 2.024)$, $\Psi _3= -0.015$ with ${\mathbb {W}}_{b}=(-0.015, 0.1945)$.

Table 14 CF between HOF and $\Re {GA}(FeTe_{2})$.

Fig. 16

HOF of $\Re {GA}(FeTe_{2})$.

• HOF using $\Re {PM_1}(FeTe_{2})$

$$\begin{aligned} f(\Re {PM_1}) = \frac{(\kappa _1\times \Re {PM_1}^4 + \kappa _2\times \Re {PM_1}^3 + \kappa _3\times \Re {PM_1}^2 + \kappa _4\times \Re {PM_1} + \kappa _5)}{(\Re {PM_1} + \Psi _1)} \end{aligned}$$

In which the normalized mean of $\Re {PM_1}$ is $4.446e+08$ and the standard deviation is $6.376e+08$. The coefficients are: $\kappa _1= 36.73$, with ${\mathbb {W}}_{b}=(-572.5, 646)$, $\kappa _2= -65.91$, with ${\mathbb {W}}_{b}=(-1156, 1024)$, , $\kappa _5= 1.853$, $\kappa _3= -31.2$ with ${\mathbb {W}}_{b}=(-551.4, 489)$, $\kappa _4= 27.52$ with ${\mathbb {W}}_{b}=(-457.4, 512.4)$ with ${\mathbb {W}}_{b}=(-44.13, 47.84)$, $\Psi _1= 1.565$ with ${\mathbb {W}}_{b}=(-16.02, 19.15)$.

Fig. 17

HOF of $\Re {PM_1}(FeTe_{2})$.

Table 15 HOF and $\Re {PM_1}(FeTe_{2})$.

• HOF using $\Re {PM_2}(FeTe_{2})$

$$\begin{aligned} f(\Re {PM_2}) = \frac{(\kappa _1\times \Re {PM_2}^4 + \kappa _2\times \Re {PM_2}^3 + \kappa _3\times \Re {PM_2}^2 + \kappa _4\times \Re {PM_2} + \kappa _5)}{(\Re {PM_2} + \Psi _1)} \end{aligned}$$

In which the normalized mean of $\Re {PM_2}$ is $1.334e+08$ and the standard deviation is $1.913e+08$. The coefficients are: $\kappa _1= 36.73$, with ${\mathbb {W}}_{b}=(-572.5, 645.9)$, $\kappa _2= -65.91$, with ${\mathbb {W}}_{b}=(-1156, 1024)$, $\kappa _3= -31.2$ with ${\mathbb {W}}_{b}=(-551.4, 489)$, $\kappa _4= 27.52$ with ${\mathbb {W}}_{b}=(-457.4, 512.4)$, $\kappa _5= 1.853$ with ${\mathbb {W}}_{b}=(-44.13, 47.84)$, $\Psi _1= 1.565$ with ${\mathbb {W}}_{b}= (-16.02, 19.15)$.

Fig. 18

HOF of $\Re {PM_2}(FeTe_{2})$.

Table 16 CF between HOF and $\Re {PM_2}(FeTe_{2})$.

• HOF using $\Re {F}(FeTe_{2})$

$$\begin{aligned} f(\Re {F}) =\frac{(\kappa _1\times \Re {F}^3 +\kappa _2\Re {F}^2+\kappa _3\times \Re {F}+\kappa _4 )}{(\Re {F}^3+\Psi _1\Re {F}^2 + \Psi _2\Re {F} + \Psi _3)} \end{aligned}$$

In which the normalized mean of $\Re {F}$ is 492 and the standard deviation is 367.3. The coefficients are: $\kappa _1= -2.439$, with ${\mathbb {W}}_{b}=(-4.702, -0.1757)$, $\Psi _1= 1.216$ with ${\mathbb {W}}_{b}=(0.5467, 1.885)$, $\kappa _2= -2.316$, with ${\mathbb {W}}_{b}=(-6.522, 1.889)$, $\kappa _3= 0.2778$ with ${\mathbb {W}}_{b}=(-1.437, 1.992)$, $\kappa _4= 0.1644$ with ${\mathbb {W}}_{b}=(-1.84, 2.169)$, $\Psi _2= 0.1702$ with ${\mathbb {W}}_{b}=(-0.2916, 0.6319)$, $\Psi _3= -0.02821$ with ${\mathbb {W}}_{b}=(-0.2722, 0.2158)$.

Table 17 CF between HOF and $\Re {F}(FeTe_{2})$.

Fig. 19

HOF of $\Re {F}(FeTe_{2})$.

Key findings and discussion

We provide a number of new reverse degree-based topological indices that are especially designed for Iron Telluride $FeTe_{2}$ networks, and provide their mathematical definitions. Reverse Randic, Reverse Balaban, and Reverse Zagreb indices are a few of the variants of these indices that are designed to more precisely represent the complex topological characteristics of $FeTe_{2}$ networks. We used a variety of fitting models, such as logarithmic, polynomial, and linear regressions, to create strong correlations between the Iron Telluride network’s important experimental characteristics and topological indices. The design and optimisation of $FeTe_{2}$-based materials will be significantly impacted by the study’s findings. Reverse degree-based topological indices provide valuable insights and effective instruments for comprehending and forecasting material characteristics.

The models showed promise as prediction tools since they showed strong relationships between the topological indices and important experimental characteristics including mechanical strength, thermal stability, and electrical conductivity. Comparing Linear and Non-Linear Models: The Reverse Randic Index’s linear model performed exceptionally well in predicting electrical conductivity, pointing to a clear link. Nonetheless, non-linear models-logarithmic for mechanical strength and polynomial for thermal stability-were more appropriate for reflecting the intricate interactions that are inherent in both qualities.

The degree and distribution of atomic bonds have a major influence on a material’s resistance to thermal degradation, as demonstrated by the link between thermal stability and the Reverse Zagreb Indices. A more stable structure is formed by a network that has a higher degree of bonding, as shown by the indices. This realisation can help direct the synthesis of materials with improved thermal stability by emphasising uniform distribution and bond strength maximisation. By maximising atomic connection and minimising structural flaws that obstruct electron flow, Iron Telluride networks with improved electrical conductivity may be designed using the knowledge gathered from the Reverse Randic Index.

Electrical conductivity and the Reverse Randic Index have a substantial association $(R2 = 0.85)$, indicating that conductivity may be greatly increased by optimising atom connectivity within the $FeTe_{2}$ network. Researchers can produce materials with better conductive qualities that may be useful for electrical and optoelectronic applications by creating $FeTe_{2}$ derivatives with higher connectivity indices. The creation of $FeTe_{2}$ derivatives for structural applications can be guided by the understanding of mechanical strength offered by the Reverse Balaban Index. These materials might find use in the automotive, aerospace, and construction sectors where strong and long-lasting qualities are necessary.

Conclusion

For ${\rm FeTe}_2$, we developed a set of indices using the reverse degree-based index approach. In our work, we added closed-form expressions and the heat of formation (HOF), together with degree-based topological indices. To completely assess these estimations, we employed graphical displays in addition to computational calculations. MATLAB was used for precise numerical calculations, and Maple facilitated the creation of educational graphics. Our research aimed to establish a complete set of descriptors for ${\rm FeTe}_2$ by exploring reverse degree-based indices. We looked at the relationships between these indices and volatility measurements using advanced curve fitting techniques and a variety of error metrics analysis. The logical method consistently proven to be the most successful, and the rational fit technique consistently demonstrated to be the most accurate in terms of mobility and indices. We now have a helpful tool for optimizing structural modifications to ${\rm FeTe}_2$ so that its properties can be better suited for a range of practical applications thanks to this project.