
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39227651
71645
10.1038/s41598-024-71645-3
Article
On statistical evaluation of reverse degree based topological indices for iron telluride networks
Youssef Maged Z. 1
Al-Dayel Ibrahim 1
Hanif Muhammad Farhan 2
Siddiqui Muhammad Kamran 3
Ahmed Hira 3
Tolasa Fikadu Tesgera fikadu@dadu.edu.et

4
1 https://ror.org/05gxjyb39 grid.440750.2 0000 0001 2243 1790 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, 11566 Riyadh, Saudi Arabia
2 https://ror.org/051jrjw38 grid.440564.7 0000 0001 0415 4232 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
3 https://ror.org/00nqqvk19 grid.418920.6 0000 0004 0607 0704 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan
4 https://ror.org/00zvn8514 0000 0005 0599 1779 Department of Mathematics, Dambi Dollo University, Oromia, Ethiopia
4 9 2024
4 9 2024
2024
14 2053329 3 2024
29 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
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.

Keywords

Topological indices
Reverse general Randic index
Reverse atomic bond connectivity index
Reverse geometric arithmetic index
Reverse Zagreb type indices
Iron telluride
Subject terms

Applied mathematics
Physical chemistry
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

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 ς is the number of edges that intersect with it and is denoted by Ξ(ς). 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 (FeTe2), is a hot topic in materials research and solid-state physics. In the context of (FeTe2), our work seeks to clarify these indices and their statistical characteristics. The reverse degree-based topological indices, which are numerical descriptors obtained from the (FeTe2) 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 (FeTe2) network and perform statistical tests to clarify relationships between these metrics and measurable physical events related to (FeTe2)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 ℜ(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 functions13,14.

Reversing the general Randi index by Milan Randi15 yields:1 ℜRρ(G)=∑ςε∈E(G)[ℜΞ(ς)×ℜΞ(ε)]ρ;ρ=1,-1,12,-12.

The reverse-atom bond connectivity index by Estrada et al.16,17is :2 ℜABC(G)=∑ςε∈E(G)ℜΞ(ς)+ℜΞ(ε)-2ℜΞ(ς)×ℜΞ(ε).

The reverse geometric arithmetic by Vukicevic et al.18,19 index is defined as:3 ℜGA(G)=∑ςε∈E(G)2ℜΞ(ς)×ℜΞ(ε)ℜΞ(ς)+ℜΞ(ε).

The reverse first and second Zagreb indices by Gutman20,21 are defined as:4 ℜM1(G)=∑ςε∈E(G)(ℜΞ(ς)+ℜΞ(ε)).

5 ℜM2(G)=∑ςε∈E(G)(ℜΞ(ς)×ℜΞ(ε)).

The reverse hyper Zagreb index by Shirdel et al.22 is:6 ℜHM(G)=∑ςε∈E(G)[ℜΞ(ς)+ℜΞ(ε)]2.

The reverse forgotten index by Furtula and Gutman23,24 is defined as:7 ℜF(G)=∑ςε∈E(G)[ℜΞ(ς)2+ℜΞ(ε)2].

The reverse first multiple and reverse second multiple Zagreb degree-based indices by Ghorbani and Azimi25 are defined as:8 ℜPM1(G)=∏ςε∈E(G)[ℜΞ(ς)+ℜΞ(ε)].

9 ℜPM2(G)=∏ςε∈E(G)[ℜΞ(ς)×ℜΞ(ε)].

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 (FeTe2) has attracted a lot of interest in the field of materials research. It is a layered material with a typical formula of MX2, 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 FeTe2 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 (FeTe2) formulae using a unit cell, whereas Figure 4 shows a more general construction. To get the topological indices of iron telluride (FeTe2), the edge partition will be considered as follows: Table 1 shows the edge partition of iron telluride (FeTe2) when (m, n) is larger than or equal to 1. Using the formulaℜΞ(ς)=Δ(G)-Ξ(ς)+1.

, where Δ(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 E1, E2, E3, E4, and E5.Fig. 3 The structure of iron telluride (FeTe2) for n=m=227.

Fig. 4 The structure of iron telluride (FeTe2) for m=2, n=327.

Results for iron telluride network (FeTe2)

In the molecular graph of FeTe2 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 FeTe2 is 4mn+2m+n+1 and 6mn+m, respectively. Table 1 displays the edge partition of FeTe2. Table 2 displays the reverse degree-based edge partition of FeTe2.Table 1 Edge partition of FeTe2 established on degrees of terminal vertices.

(Ξ(ς), Ξ(ε))	Frequency	
(1, 2)	2	
(1, 3)	n+1	
(2, 2)	2(m-1)	
(2, 3)	2(-2+2m+n)	
(3, 3)	3(2mn-n+1)-5m	

Table 2 Reverse degree based edge partition of FeTe2.

(ℜΞ(ς),ℜΞ(ε))	Frequency	
(3, 2)	2	
(3, 1)	n+1	
(2, 2)	(2(m-1))	
(2, 1)	2(-2+2m+n)	
(1, 1)	3(2mn-n+1)-5m	

Reverse general Randic index

By considering the Eq. (1) and Table 2, we make computation as below:ℜRρ(FeTe2)=∑ςε∈E(FeTe2)[ℜΞ(ς)×ℜΞ(ε)]ρ;ρ=1,-1,12,-12.

Forρ=1x;ℜR1(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)[ℜΞ(ς)×ℜΞ(ε)]=(3×2)(2)+(3×1)(n+1)+(2×2)((2(m-1)))+(2×1)(2(-2+2m+n))+(1×1)(3(2mn-n+1)-5m)ℜR1(FeTe2)=6mn+11m+4n+2.

Forρ=-1;ℜR-1(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)1[ℜΞ(ς)×ℜΞ(ε)]=23×2+n+13×1+(2(m-1))2×2+2(-2+2m+n)2×1+3(2mn-n+1)-5m1×1ℜR-1(FeTe2)=6mn-2.5m-1.666667n+1.166667.

Forρ=12;ℜR12(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)ℜΞ(ς)×ℜΞ(ε)=(2)3×2+(n+1)3×1+((2(m-1)))2×2+(2(-2+2m+n))2×1+(3(2mn-n+1)-5m)1×1ℜR12(FeTe2)=6mn+4.656854m+1.560478n-0.025824.

Forρ=-12;ℜR-12(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)1ℜΞ(ς)×ℜΞ(ε)=23×2+n+13×1+(2(m-1))2×2+2(-2+2m+n)2×1+3(2mn-n+1)-5m1×1ℜR-12(FeTe2)=6mn-1.171573m-1.008436n+0.56542.

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 ℜR1(FeTe2), ℜR-1(FeTe2), ℜR12(FeTe2), ℜR-12(FeTe2).

[m, n]	ℜR1(FeTe2)	ℜR-1(FeTe2)	ℜR12(FeTe2)	ℜR-12(FeTe2)	
[1, 1]	23	3	12.191508	4.385411	
[2, 2]	56	16.833333	36.40884	20.205402	
[3, 3]	101	42.666666	72.626172	48.025393	
[4, 4]	158	80.499999	120.843504	87.845384	
[5, 5]	227	130.333332	181.060836	139.665375	
[6, 6]	308	192.166665	253.278168	203.485366	
[7, 7]	401	265.999998	337.4955	279.305357	
[8, 8]	506	351.833331	433.712832	367.125348	

Fig. 5 Comparison graph of ℜR1(FeTe2), ℜR-1(FeTe2), ℜR12(FeTe2), ℜR-12(FeTe2)  (https://www.originlab.com/).

Reverse atom bond connectivity index

By considering Eq. (2) and Table 2, we make computation as below:ℜABC(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)ℜΞ(ς)+ℜΞ(ε)-2ℜΞ(ς)×ℜΞ(ε)=(2)3+2-23×2+(n+1)3+1-23×1((2(m-1)))+2+2-22×2+(2(-2+2m+n))2+1-22×1+(3(2mn-n+1)-5m)1+1-21×1ℜABC(FeTe2)=4.242641m+2.23071n-2.011931.

Reverse geometric arithmetic index

By considering the Eq. (3) and Table 2, we make computation as below:ℜGA(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)2ℜΞ(ς)×ℜΞ(ε)ℜΞ(ς)+ℜΞ(ε)=23×23+2(2)+23×13+1(n+1)+22×22+2((2(m-1)))+22×12+1(2(-2+2m+n))+21×11+1(3(2mn-n+1)-5m)ℜGA(FeTe2)=6mn+0.771236m-0.248357n+0.054381.

Reverse first Zagreb index

By considering the Eq. (4) and Table 2, we make computation as below:ℜM1(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)(ℜΞ(ς)+ℜΞ(ε))=(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)ℜM1(FeTe2)=12mn+10m+4n.

Reverse second Zagreb index

By considering the Eq. (5) and Table 2, we make computation as below:ℜM2(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)(ℜΞ(ς)×ℜΞ(ε))=(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)ℜM2(FeTe2)=6mn+11m+4n+2.

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 ℜABC(FeTe2), ℜGA(FeTe2), ℜM1(FeTe2), ℜM2(FeTe2).

[m, n]	ℜABC(FeTe2)	ℜGA(FeTe2)	ℜM1(FeTe2)	ℜM2(FeTe2)	
[1, 1]	4.46142	6.57726	26	23	
[2, 2]	10.934771	25.100139	76	56	
[3, 3]	17.408122	55.623018	150	101	
[4, 4]	23.881473	98.145897	248	158	
[5, 5]	30.354824	152.668776	370	227	
[6, 6]	36.828175	219.191655	516	308	
[7, 7]	43.301526	297.714534	686	401	
[8, 8]	49.774877	388.237413	880	506	

Fig. 6 Comparison graph of ℜABC(FeTe2), ℜGA(FeTe2), ℜM1(FeTe2), ℜM2(FeTe2)  (https://www.originlab.com/).

Reverse hyper Zagreb index

By considering the Eq. (6) and Table 2, we make computation as below:ℜHM(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)[ℜΞ(ς)+ℜΞ(ε)]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)2ℜHM(FeTe2)=24mn+48m+22n+10.

Reverse First Multiple Zagreb Index

By considering the Eq. (8) and Table 2, we make computation as below:ℜPM1(FeTe2)=∏i=15∏ςε∈Ei(FeTe2)[ℜΞ(ς)+ℜΞ(ε)]=(2)(3+2)×(n+1)(3+1)×(2(-2+2m+n))(2+1)×((2(m-1)))(2+2)×(3(2mn-n+1)-5m)(1+1)ℜPM1(FeTe2)=46080m3n2+23040m2n3+7680m3n-111360m2n2-34560mn3-38400m3-34560m2n+88320mn2+11520n3+99840m2+38400mn-23040n2-84480m-11520n+23040.

Reverse second multiple Zagreb index

By considering the Eq. (9) and Table 2, we make computation as below:ℜPM2(FeTe2)=∏i=15∏ςε∈Ei(FeTe2)[ℜΞ(ς)×ℜΞ(ε)]=(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)ℜPM2(FeTe2)=13824m3n2+6912m2n3+2304m3n-33408m2n2-10368mn3-11520m3-10368m2n+26496mn2+3456n3+29952m2+11520mn-6912n2-25344m-3456n+6912.

Reverse forgotten index

By considering the Eq. (7) and Table 2, we make computation as below:ℜF(FeTe2)=∑i=15∑ςε∈Ei(FeTe2)[ℜΞ(ς)2+ℜΞ(ε)2]=(2)(32+22)+(n+1)(32+12)+((2(m-1)))(22+22)++(2(-2+2m+n))(22+12)+(3(2mn-n+1)-5m)(12+12)ℜF(FeTe2)=12mn+26m+14n+6.

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 ℜHM(FeTe2), ℜPM1(FeTe2), ℜPM2(FeTe2).

[m, n]	ℜHM(FeTe2)	ℜPM1(FeTe2)	ℜPM2(FeTe2)	ℜF(FeTe2)	
[1, 1]	104	0	0	58	
[2, 2]	246	506880	152064	134	
[3, 3]	436	7096320	2128896	234	
[4, 4]	674	38592000	11577600	358	
[5, 5]	960	135383040	40614912	506	
[6, 6]	1294	367718400	110315520	678	
[7, 7]	1676	844001280	253200384	874	
[8, 8]	2106	1719083520	515725056	1094	

Fig. 7 Comparison graph of ℜHM(FeTe2), ℜPM1(FeTe2), ℜPM2(FeTe2), ℜF(FeTe2)   (https://www.originlab.com/).

Rational curve fitting (Rcfmn) 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 (FeTe2). Then, the HoF of Iron Telluride (FeTe2) 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 ‘δHf’ 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 FeTe2. Divide FeTe2=-51.9/-65.8kJ/mol by Avogadro’s number, 6.02214×1023mol-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, \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 R2-test. A good match is shown by R2 nearing 1, whereas a poor estimate is indicated by R2 approaching 0. The ratio of estimated variance to actual variance is denoted by R2. 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 RMSE, SSE, r2, and adj(r2), to evaluate the explanatory power and accuracy of the fitted models.HOF using ℜR1(FeTe2)

f(ℜR1)=(χ1×ℜR13+χ2×ℜR12+χ3×ℜR1+χ4)(ℜR13+Ψ1×ℜR12+Ψ2×ℜR1+Ψ3)

In which the Normalized mean of ℜR1 is 222.5 and the standard deviation is 171.6. The coefficients are: κ1=-2.431, with Cb=(-4.717,-0.1442), Ψ1=1.233 with Wb=(0.5785,1.889), Ψ2=0.196 with Wb=(-0.2597,0.6518), χ2=-2.356, with Wb=(-6.567,1.854), κ3=0.2201 with Wb=(-1.498,1.938), κ4=0.1682,with Wb=(-1.803,2.14), Ψ3=-0.026 with Wb=(-0.2623,0.2103).Fig. 8 HOF of ℜR1(FeTe2).

Table 6 HOF and ℜR1(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04834	0.9989	0.9925	0.2199	

HOF using ℜR-1(FeTe2)

f(ℜR-1)=(κ1×ℜR-13+κ2×ℜR-12+κ3×ℜR-1+κ4)(ℜR-13+Ψ1×ℜR-12+Ψ2×ℜR-1+Ψ3)

In which Normalized mean of ℜR-1 is 135.4 and standard deviation is 125.6. The coefficients are: κ1=-2.385, with Wb=(-4.822,0.05173), κ2=-2.581, with Wb=(-6.816,1.655), Ψ1=1.33 with Wb=(0.7634,1.896), κ3=-0.1022, with Wb=(-1.848,1.643),κ4=0.1776, with Wb=(-1.616,1.971), Ψ2=0.34 with Wb=(-0.07087,0.7509), Ψ3=-0.008958 with Wb=(-0.2082,0.1903).Table 7 CF between HOF and ℜR-1(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.05077	0.9989	0.9921	0.2253	

Fig. 9 HOF of ℜR-1(FeTe2).

HOF using ℜR12(FeTe2)

f(ℜR12)=(κ1×ℜR123+κ2×ℜR122+κ3×ℜR12+κ4)(ℜR123+Ψ1×ℜR122+Ψ2×ℜR12+Ψ3)

In which normalized mean of ℜR12 is 181 and standard deviation is 150.4. The coefficients are: κ1=-2.413, with Wb=(-4.752,-0.07324), κ2=-2.442, with Ψ1=1.27 with Wb=(0.6477,1.893), Ψ2=0.2513 with Wb=(-0.1896,0.6922), Wb=(-6.663,1.779), κ3=0.09659 with Wb=(-1.63,1.823), κ4=0.1742 with Wb=(-1.728,2.076), Ψ3=-0.0204 with Wb=(-0.2414,0.2006).Fig. 10 HOF of ℜR12(FeTe2).

Table 8 CF between HOF and ℜR12(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.0492	0.9989	0.9923	0.2218	

HOF using ℜR-12(FeTe2)

f(ℜR-12)=(κ1×ℜR-123+κ2×ℜR-122+κ3×ℜR-12+κ4)(ℜR-123+Ψ1×ℜR-122+Ψ2×ℜR-12+Ψ3)

Fig. 11 HOF of ℜR-12(FeTe2).

In which Normalized mean of ℜR-12 is 143.8 and standard deviation is 130.3. The coefficients are: κ1=-2.391, with Wb=(-4.805,0.02294), κ2=-2.55 with Ψ1=1.317 with Wb=(0.7374,1.896), Wb=(-6.783,1.683), κ3=-0.05842, with Wb=(-1.799,1.682), κ4=0.1775, with Wb=(-1.64,1.995), Ψ2=0.3205 with Wb=(-0.09767,0.7386), Ψ3=-0.01173 with Wb=(-0.2154,0.192).Table 9 HOF and ℜR-12(FeTe2).

Fir-type	SSE	r2	adj(r2)	RMSE	
r33	0.0504	0.9989	0.9921	0.2245	

HOF using ℜM1(FeTe2)

Fig. 12 HOF of ℜM1(FeTe2).

f(ℜM1)=(κ1×ℜM13+κ2×ℜM12+κ3×ℜM1+κ4)(ℜM13+Ψ1×ℜM12+Ψ2×ℜM1+Ψ3)

In which Normalized mean of ℜM1 is 369 and standard deviation is 304.6. The coefficients are: κ1=-2.415, with Wb=(-4.749,-0.08057), κ2=-2.434, with Wb=(-6.653,1.786), Ψ1=1.267 with Wb=(0.6407,1.892), κ3=0.109 with Wb=(-1.617,1.835), κ4=0.1737, with Wb=(-1.735,2.083), Ψ2=0.2458 with Wb=(-0.1968,0.6883), Ψ3=-0.02102 with Wb=(-0.2435,0.2014).Table 10 HOF and ℜM1(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04911	0.9989	0.9923	0.2216	

HOF using ℜM2(FeTe2)

f(ℜM2)=(κ1×ℜM23+κ2×ℜM22+κ3×ℜM2+κ4)(ℜM23+Ψ1×ℜM22+Ψ2×ℜM2+Ψ3)

In which the Normalized mean of ℜM2 is 222.5 and the standard deviation is 171.6. The coefficients are: κ1=-2.431, with Wb=(-4.717,--0.1442), κ2=-2.356, Ψ1=1.233 with Wb=(-6.567,1.854), κ3=0.2201 with Wb=(-1.498,1.938), κ4=0.1682 with Wb=(-1.803,2.14), with Wb=(0.5785,1.888), Ψ2=0.196 with Wb=(-0.2597,0.6518), Ψ3=-0.026 with Wb=(-0.2623,0.2103).Fig. 13 HOF of ℜM2(FeTe2).

Table 11 CF between HOF and ℜM2(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04834	0.9989	0.9925	0.2199	

HOF using ℜHM(FeTe2)

f(ℜHM)=(κ1×ℜHM3+κ2×ℜHM2+κ3×ℜHM+κ4)(ℜHM3+Ψ1ℜHM2+Ψ2ℜHM+Ψ3)

In which the normalized mean of ℜHM is 937 and the standard deviation is 710.4. The coefficients are: κ1=-2.435, with Wb=(-4.709,-0.1606), Ψ1=1.224 with Wb=(0.562,1.887), Ψ2=0.1827 with Wb=(-0.2763,0.6416),κ2=-2.336, with Wb=(-6.543,1.872), κ3=0.2499, with Wb=(-1.466,1.966), κ4=0.1663, with Wb=(-1.822,2.155), Ψ3=-0.02718 with Wb=(-0.2674,0.2131).Fig. 14 HOF of ℜHM(FeTe2).

Table 12 CF between HOF and ℜHM(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04815	0.9989	0.9925	0.2194	

HOF using ℜABC(FeTe2)

f(ℜABC)=(κ1×ℜABC5+κ2×ℜABC4+κ3×ℜABC3+κ4×ℜABC2+κ5×ℜABC+κ6)(ℜABC+Ψ1)

Fig. 15 HOF of ℜABC(FeTe2).

In which the normalized mean of ℜABC is 27.12 and the standard deviation is 15.86. The coefficients are: κ1=-2.051, with Wb=(-2.489,-1.614), κ2=1.785 with Wb=(1.312,2.258), κ5=-4.27 with Wb=(-4.736,-3.805), κ6=2.654 with Wb=(2.254,3.054), κ=34.811 with Wb=(3.783,5.839), κ4=-3.963 with Wb=(-4.964,-2.962), Ψ1=-0.3968, with Wb=(-0.4378,-0.3559).Table 13 CF between HOF and ℜABC(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r51	0.001144	1	0.9998	0.03382	

HOF using ℜGA(FeTe2)

f(ℜGA)=(κ1×ℜGA3+κ2×ℜGA2+κ3×ℜGA+κ4)(ℜGA3+Ψ1×ℜGA2+Ψ2×ℜGA+Ψ3)

In which the normalized mean of ℜGA is 155.4 and the standard deviation is 136.7. The coefficients are: κ1=-2.399, with Wb=(-4.785,-0.01203), κ2=-2.512, with Wb=(-6.741,1.1717), Ψ1=1.3 with Wb=(0.7053,1.895), Ψ2=0.2961, with Wb=(-0.1307,0.7228),κ3=-0.00363, with Wb=(-1.739,1.731), κ4=0.1768, with Wb=(-1.67,2.024), Ψ3=-0.015 with Wb=(-0.015,0.1945).Table 14 CF between HOF and ℜGA(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04996	0.9989	0.9922	0.2235	

Fig. 16 HOF of ℜGA(FeTe2).

HOF using ℜPM1(FeTe2)

f(ℜPM1)=(κ1×ℜPM14+κ2×ℜPM13+κ3×ℜPM12+κ4×ℜPM1+κ5)(ℜPM1+Ψ1)

In which the normalized mean of ℜPM1 is 4.446e+08 and the standard deviation is 6.376e+08. The coefficients are: κ1=36.73, with Wb=(-572.5,646), κ2=-65.91, with Wb=(-1156,1024), , κ5=1.853, κ3=-31.2 with Wb=(-551.4,489), κ4=27.52 with Wb=(-457.4,512.4) with Wb=(-44.13,47.84), Ψ1=1.565 with Wb=(-16.02,19.15).Fig. 17 HOF of ℜPM1(FeTe2).

Table 15 HOF and ℜPM1(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r41	0.4302	0.9904	0.9425	0.6559	

HOF using ℜPM2(FeTe2)

f(ℜPM2)=(κ1×ℜPM24+κ2×ℜPM23+κ3×ℜPM22+κ4×ℜPM2+κ5)(ℜPM2+Ψ1)

In which the normalized mean of ℜPM2 is 1.334e+08 and the standard deviation is 1.913e+08. The coefficients are: κ1=36.73, with Wb=(-572.5,645.9), κ2=-65.91, with Wb=(-1156,1024), κ3=-31.2 with Wb=(-551.4,489), κ4=27.52 with Wb=(-457.4,512.4), κ5=1.853 with Wb=(-44.13,47.84), Ψ1=1.565 with Wb=(-16.02,19.15).Fig. 18 HOF of ℜPM2(FeTe2).

Table 16 CF between HOF and ℜPM2(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r41	0.4302	0.9904	0.9425	0.6559	

HOF using ℜF(FeTe2)

f(ℜF)=(κ1×ℜF3+κ2ℜF2+κ3×ℜF+κ4)(ℜF3+Ψ1ℜF2+Ψ2ℜF+Ψ3)

In which the normalized mean of ℜF is 492 and the standard deviation is 367.3. The coefficients are: κ1=-2.439, with Wb=(-4.702,-0.1757), Ψ1=1.216 with Wb=(0.5467,1.885), κ2=-2.316, with Wb=(-6.522,1.889), κ3=0.2778 with Wb=(-1.437,1.992), κ4=0.1644 with Wb=(-1.84,2.169), Ψ2=0.1702 with Wb=(-0.2916,0.6319), Ψ3=-0.02821 with Wb=(-0.2722,0.2158).Table 17 CF between HOF and ℜF(FeTe2).

Fit-type	SSE	r2	adj(r2)	RMSE	
r33	0.04797	0.9989	0.9925	0.219	

Fig. 19 HOF of ℜF(FeTe2).

Key findings and discussion

We provide a number of new reverse degree-based topological indices that are especially designed for Iron Telluride FeTe2 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 FeTe2 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 FeTe2-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 FeTe2 network. Researchers can produce materials with better conductive qualities that may be useful for electrical and optoelectronic applications by creating FeTe2 derivatives with higher connectivity indices. The creation of FeTe2 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 FeTe2, 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 FeTe2 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 FeTe2 so that its properties can be better suited for a range of practical applications thanks to this project.

Acknowledgements

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RPP2023042).

Author contributions

Data analysis, computation, financing resources, and calculation verifications were aided by Maged Z. Youssef. Ibrahim Al-Dayel examined and approved the paper’s final text in addition to providing computational support. Muhammad Farhan Hanif makes contributions to the advancement of Maple graphs and Matlab computations. Muhammad Kamran Siddiqui prepared the first draft of the paper and assisted with project management, conception, methodology, supervision, and resource gathering. Hira Ahmed helped with the investigation, experiment design, and data curation analysis. Funding, software, validation, and formal analysis of experiments are all aided by Fikadu Tesgera Tolasa. The final draft was read and approved by all writers.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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