Analyzing Topological Indices and Heat of Formation for Copper(II) Fluoride Network via Curve Fitting Models

ABSTRACT

One of the most important fields in current materials research is the examination of materials through the prism of topological indices. An extensive statistical investigation of topological indices relevant to the characterization of copper(II) fluoride is presented in this paper. Our goal is to use the well-understood structural features of copper(II) fluoride, a chemical that is well known for its crystalline qualities, to uncover a variety of properties inherent in its network. This work makes use of the structural information available for copper(II) fluoride. After a thorough computational investigation, a rigorous statistical analysis is conducted to determine the distributions and correlations between heat of formation and the different topological indices. The findings reveal significant patterns and trends in the copper(II) fluoride network structure, providing insights into the underlying principles governing its material behavior.

Introduction

A subfield of theoretical chemistry and mathematics known as “chemical graph theory” is concerned with the application of graph theoretical ideas to the study of molecular structures. In this sense, atoms are the vertices of a graph, and chemical bonds are the edges that connect the vertices, representing a molecule. A graph is defined as an ordered pair $\mathit{G}=(\mathit{V},\mathit{E})$, where V is a set of vertices and E is a set of edges (Wagner and Wang 2018; West 2001). It gives an estimate of the number of vertices in the graph, and its cardinality is called order of graph. The total number of edges in a graph determines its size (Zhang et al. 2018). The number of edges incident on a vertex in a graph is called its degree and is represented by $\mathcal{\S }(\mathit{\varphi })$.

Numerous domains, such as biology, social media, computer networks, and logistics, have applications in graph theory (Siddiqui et al. 2016). In computer networks, graphs are used to optimize data transfer by simulating the links between devices. In social networks, graphs are used to analyze user relationships. In logistics, graphs are used to optimize transportation routes. In biology, they behave as molecular structures that facilitate the study of protein interactions and genetic connections (Siddiqui, Imran, and Ahmad 2016; Zhang et al. 2022).

The depiction of molecular structures using graph-based models is the focus of chemical graph theory. Atoms are represented as vertices and chemical bonds as edges (Zhang et al. 2018). This model facilitates the examination of molecular properties, reaction mechanisms, and predictions of chemical behavior. Chemical graph theory plays a crucial role in drug development by providing insight into molecular structures. By anticipating the physical and chemical characteristics of substances, it helps in the creation of new drugs. Moreover, chemical graph theory is used in computational chemistry to analyze and simulate complex chemical reactions (Zhang et al. 2021).

Liu et al. (2021) and Liu, Bao, and Zheng (2022) analyzed some structural properties of networks. With the use of these methods, scientists can examine chemical graphs more thoroughly and gain a more complex knowledge of their topological characteristics (Ali Malik et al. 2023; Koam, Ahmad, and Nadeem 2021). Ahmad et al. (2020, 2021) conducted a theoretical study of energy of phenylene and anthracene. These techniques have helped to enhance chemistry and enable more precise manipulation of molecular structures. Predicting chemical behaviors aids in the development of innovative pharmaceuticals, which makes this precision particularly helpful in the drug-discovery process. Chemical graph theory is fundamentally a basis for expanding the frontiers of chemical studies and eventually contributing to the identification of novel therapeutics and remedies. It offers an efficient set of ideas and procedures (Zhang et al. 2022).

Topological indices are numerical measures associated with the structural characteristics of a graph that provide significant information on the topological properties of the graph (Nadeem et al. 2021). The branching patterns, cyclic structures, and connectivity of the graph are described in detail by these numerical parameters, also referred to as indices. By looking at these indices, researchers can discover more about the fundamental topological properties of chemical graphs. Moreover, topological indices are crucial for researching the quantitative structure–activity relationship (QSAR), which is the link between a compound’s chemical structure and biological activity.

A degree topological index with a high degree of selectivity was studied by Balaban (1982). Estrada et al. (1998) focused on an index called atom-bond connectivity, whereas Furtula, Graovac, and Vukievi (2010) investigated the augmented Zagreb index and furthermore looked at a forgotten topological index. Multiple Zagreb indexes were briefly reviewed by Ghorbani and Azimi (2012). Shirdel, Rezapour, and Sayadi (2013) studied the hyper-Zagreb index of graph operations. Zaman et al. (2024) discussed the neighborhood topological indices of nanostructures. Ullah et al. (2024) analyzed the Zagreb-type molecular descriptors for some novel drug molecules. Zaman et al. (2023) provided some novel topological indices for graph networks. Ullah et al. (2023) computed irregularity topological indices for some bioconjugate networks. Sharma, Bhat, and Sharma (2022) discussed the topological indices of carbon nanocones. Aqib et al. (2023) analyzed the topological indices of some chemical graphs. Sharma, Bhat, and Liu (2023) provided second leap hyper-Zagreb coindex of certain benzenoid structures.

The scientometric analysis of the degree-based topological indices by different countries worldwide is displayed in Figure 1.

Figure 1. Scientometric analysis: topological indices based on degree, across different nations.

We have presented the scientometric study of the degree-based topological indices keywords in multiple ways in Figure 2. This study shows that there are many different methods in which degree-based topological indices keywords are addressed.

Figure 2. Scientometric analysis: keywords for topological indices based on degree.

Structure of Copper(II) Fluoride$(\mathit{Cu}{F}_2)$

Copper(II) fluoride is also known as cupric fluoride or copper (2+) difluoride. Copper(II) fluoride is a chemical made up of copper in the +2 oxidation state and fluorine ions. Its molecular structure is mostly dictated by how these ions are arranged in the crystal lattice. In the solid state, copper(II) fluoride has a crystalline structure that is commonly monoclinic or tetragonal. Cu(II) ions (Cu2+) are transition metal cations that are surrounded by fluoride ions (F-) in a precise geometric configuration to achieve electrostatic stability (Fischer et al. 1974). In the crystal lattice of copper(II) fluoride, each copper ion is surrounded by four fluoride ions, forming a tetrahedral coordination geometry. The copper(II) ions serve as the core metal atoms, while the fluoride ions act as ligands, coordinating to the copper ions through their lone pairs of electrons. The electrostatic forces of attraction between positively charged copper ions and negatively charged fluoride ions help to maintain the overall stability of the crystal structure.

The solubility of copper(II) fluoride in water is rather low, and its color might vary based on many factors and impurities. This material is widely used in many different applications, such as serving as a precursor in the synthesis of other compounds containing copper or as a catalyst in specific chemical reactions. The cardinality of vertices and edges of $\mathit{Cu}{F}_2$ are $8\mathit{mn}+2\mathit{m}+2\mathit{n}$ and $12\mathit{mn}$. The number of vertices of degree 1 are $2\mathit{m}+2\mathit{n}+2$, number of vertices of degree 2 are $2\mathit{m}+2\mathit{n}-4$, and number of vertices of degree 3 are $8\mathit{mn}-2\mathit{m}-2\mathit{n}+2$. The edge partition for $\mathit{Cu}{F}_2$ is shown in Table 1. For unit and crystal structure of $\mathit{Cu}{F}_2$, see Figure 3.

Figure 3. (a) Unit structure of copper(II) fluoride $(\mathit{Cu}{F}_2)$ and (b) crystal sheet of copper(II) fluoride $(\mathit{Cu}{F}_2)$.

Table 1. Edge partition of $\mathit{u}{F}_2$.

$(\mathcal{\S }(\mathit{\varphi }),\mathcal{\S }(\mathit{\psi }))$	Frequency	Set of edges
(1,3)	$2\mathit{mn}+2\mathit{n}+2$	$E_1$
(2,3)	$4\mathit{m}+4\mathit{n}-8$	$E_2$
(3,3)	$12\mathit{mn}-6\mathit{m}-6\mathit{n}+6$	$E_3$

Results for Copper(II) Fluoride$\left(\mathit{Cu}{F}_2\right)$

The Randić index, introduced by Milan Randić (Randić 1975), was later generalized by Amić et al. (1998) and Bollobás and Erdös (1998) as follows:

(1) $R_{\alpha }(\mathit{G})={\displaystyle \sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\S }(\mathit{\varphi })\times \S (\mathit{\psi }){)}^{\alpha },\text{Where\hspace{0.17em}}\mathit{\alpha }\text{\hspace{0.17em}=1},-\text{1},\text{\hspace{0.17em}12},-\text{12}$(1)

The computation of Randić index for $\mathit{\alpha }=1,-1,\frac{1}{2},\frac{-1}{2}$ is as follows:

$R_{\alpha }(\mathit{Cu}{F}_2)=\sum _{\varphi \psi \in E(Cu{F}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }){)}^{\alpha };\mathit{\alpha }=1,-1,\frac{1}{2},\frac{-1}{2}$

$\mathbf{For}\mathit{\alpha }=1;$

$R_1(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }){)}^1$

$=(6)(\mathit{m}+\mathit{n}+1)+(24)(\mathit{m}+\mathit{n}-2)+(54)$

$(2\mathit{mn}-\mathit{m}-\mathit{n}+1)=108\mathit{mn}-24\mathit{m}-24\mathit{n}+12.$

$\mathbf{For}\mathit{\alpha }=-1;$

$R_{-1}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }){)}^{-1}$

$=\frac{2}{(1\times 3)}(\mathit{m}+\mathit{n}+1)+\frac{1}{(2\times 3)}(4\mathit{m}+4\mathit{n}-8)+\frac{1}{(3\times 3)}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=\frac{4\mathit{m}}{3}+\frac{2\mathit{m}}{3}+\frac{2\mathit{n}}{3}.$

$\mathit{For}\mathit{\alpha }=\frac{1}{2};$

$R_{\frac{1}{2}}={\displaystyle \sum _{\mathit{i}=1}^3{\displaystyle \sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}{(\S (\mathit{\varphi })\times \S (\mathit{\psi }))}^{\frac{1}{2}}}}$

$=\sqrt{(1\times 3)}(2\mathit{m}+2\mathit{n}+2)+\sqrt{2\times 3}(4\mathit{m}+4\mathit{n}-8)+\sqrt{(3\times 3)}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=36\mathit{mn}-4.737939\mathit{m}-4.737939\mathit{n}+\mathrm{1.868184.}$

$\mathbf{For}\mathit{\alpha }=-\frac{1}{2};$

$R_{-\frac{1}{2}}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}\frac{1}{\sqrt{(\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi ))}}$

$=\frac{1}{\sqrt{(1\times 3)}}(2\mathit{m}+2\mathit{n}+2)+\frac{1}{\sqrt{2\times 3}}(4\mathit{m}+4\mathit{n}-8)+\frac{1}{\sqrt{(3\times 3)}}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=4\mathit{mn}+0.787694\mathit{m}+0.787694\mathit{n}-\mathrm{0.111286.}$

In order to quantify the degree of complexity or branching in the molecular structures depicted in Figure 4, the Randić index can be expressed graphically as a mathematical term. Furthermore included in Table 2 is a numerical analysis of Randić indices. A molecular compound’s topology is measured by a topological index in graph theory.

Figure 4. Graphical depiction of $R_1(\mathit{Cu}{F}_2)$,$R_{-1}(\mathit{Cu}{F}_2)$, $R_{\frac{1}{2}}(\mathit{Cu}{F}_2)$, and $R_{\frac{-1}{2}}(\mathit{Cu}{F}_2)$.

Table 2. Numerical comparative analysis for $R_1(\mathit{Cu}{F}_2)$, $R_{-1}(\mathit{Cu}{F}_2)$, $R_{\frac{1}{2}}(\mathit{Cu}{F}_2)$, and $R_{\frac{-1}{2}}(\mathit{Cu}{F}_2)$.

$[\mathit{m},\mathit{n}]$	$R_1(\mathit{Cu}{F}_2)$	$R_{-1}(\mathit{Cu}{F}_2)$	$R_{\frac{1}{2}}(\mathit{Cu}{F}_2)$	$R_{\frac{-1}{2}}(\mathit{Cu}{F}_2)$
(Ahmad et al. 2021)	72	2.6667	28.3923	5.461
(Zaheer et al. 2020)	348	8	126.9164	19.0395
(Ali Malik et al. 2023)	840	16	297.4405	40.6149
(Ami et al. 1998)	1548	26.6667	539.9647	70.1903
(Balaban 1982)	2472	40	854.4888	107.7657
(Balaban and Quintas 1983)	3612	56	1241.0129	153.341
(Bollobs and Erds 1998)	4968	74.6667	1699.5370	206.9164
(Das and Gutman 2004)	6540	96	2230.0612	268.4918

The $\mathit{ABC}$ index was defined by Estrada et al. (1998) and Gao et al. (2016) in the following manner:

(2) $\mathit{ABC}(\mathit{G})=\sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}\sqrt{\frac{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )-2}{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}.$(2)

The computation of atom bomb connectivity $(\mathit{ABC})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{ABC}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}\sqrt{\frac{(\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )-2)}{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}$

$=\sqrt{\frac{1}{(2)}}(2\mathit{m}+2\mathit{n}+2)+\sqrt{\frac{1}{(3)}}(4\mathit{m}+4\mathit{n}-8)+\sqrt{\frac{4}{(9)}}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=8\mathit{mn}+0.46142\mathit{m}+0.46142\mathit{n}-\mathrm{0.023861.}$

Vukievi and Furtula (2009) defined the $\mathit{GA}$ index as follows:

(3) $\mathit{GA}(\mathit{G})=\sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}\frac{2\sqrt{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}.$(3)

The computation of geometric arithmetic $(\mathit{GA})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{GA}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}\frac{2\sqrt{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}$

$=\frac{4\sqrt{1\times 3}}{(1+3)}(\mathit{m}+\mathit{n}+1)+\frac{8\sqrt{2\times 3}}{(2+3)}(\mathit{m}+\mathit{n}-2)+\frac{2\sqrt{3\times 3}}{(3+3)}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=12\mathit{mn}-0.348766\mathit{m}-0.348766\mathit{n}-\mathrm{0.106316.}$

The first and second Zagreb indices were introduced mathematically in 1972 by Gutman’s (Das and Gutman 2004; Gutman and Ch Das 2004; Gutman and Trinajsti 1972) as follows:

(4) $M_1(\mathit{G})=\sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi })).$(4)

(5) $M_2(\mathit{G})=\sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi })).$(5)

The computation of first Zagreb $(M_1)$ index for $\mathit{Cu}{F}_2$ is as follows:

$M_1(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi }))$

$=(8)(\mathit{m}+\mathit{n}+1)+(24)(\mathit{m}+\mathit{n}-2)+(36)$

$(2\mathit{mn}-\mathit{m}-\mathit{n}+1)=72\mathit{mn}-8\mathit{m}-8\mathit{n}+4.$

The computation of second Zagreb$(M_2)$ index for $\mathit{Cu}{F}_2$ is as follows:

$M_2(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }))$

$=(6)(\mathit{m}+\mathit{n}+1)+(24)(\mathit{m}+\mathit{n}-2)+(54)$

$(2\mathit{mn}-\mathit{m}-\mathit{n}+1)=108\mathit{mn}-24\mathit{m}-24\mathit{n}+12.$

The numerical comparison and graphical depiction of $\mathit{ABC}(\mathit{Cu}{F}_2)$, $\mathit{GA}(\mathit{Cu}{F}_2)$, $M_1(\mathit{Cu}{F}_2)$, and $M_2(\mathit{Cu}{F}_2)$, respectively, are shown in Table 3 and Figure 5.

Figure 5. Graphical depiction of $\mathit{ABC}(\mathit{Cu}{F}_2)$, $\mathit{GA}(\mathit{Cu}{F}_2)$, $M_1(\mathit{Cu}{F}_2)$, and $M_2(\mathit{Cu}{F}_2)$

Table 3. Numerical comparative analysis of $\mathit{ABC}(\mathit{Cu}{F}_2)$, $\mathit{GA}(\mathit{Cu}{F}_2)$, $M_1(\mathit{Cu}{F}_2)$, and $M_2(\mathit{Cu}{F}_2)$.

$[\mathit{m},\mathit{n}]$	$\mathit{ABC}(\mathit{Cu}{F}_2)$	$\mathit{GA}(\mathit{Cu}{F}_2)$	$M_1(\mathit{Cu}{F}_2)$	$M_2(\mathit{Cu}{F}_2)$
(Ahmad et al. 2021, 2021)	8.898979	11.196152	60	72
(Zaheer et al. 2020, 2020)	33.82182	46.498621	260	348
(Ali Malik et al. 2023, 2023)	74.744661	105.80109	604	840
(Ami et al. 1998, 1998)	131.667501	189.103559	1092	1548
(Balaban 1982, 1982)	204.590342	296.406028	1724	2472
(Balaban and Quintas 1983, 1983)	293.513182	427.708496	2500	3612
(Bollobs and Erds 1998, 1998)	398.436023	583.010965	3420	4968
(Das and Gutman 2004, 2004)	519.358863	762.313434	4484	6540

Shirdel, Rezapour, and Sayadi (2013) introduced the hyper-Zagreb index as

(6) $!\mathit{HM}(\mathit{G})=\sum _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi }){)}^2.$(6)

The computation of hyper-Zagreb$(\mathit{HM})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{HM}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi }){)}^2$

$=(32)(\mathit{m}+\mathit{n}+1)+(100)(\mathit{m}+\mathit{n}-2)+(81)$

$(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)=432\mathit{mn}-84\mathit{m}-84\mathit{n}+48.$

In $2012$, Ghorbani and Azimi [17] presented the first and second multiple Zagreb indices, which are as follows:

(7) $\mathit{P}{M}_1(\mathit{G})=\prod _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi })).$(7)

(8) $\mathit{P}{M}_2(\mathit{G})=\prod _{\mathit{\varphi \psi }\in \mathit{E}(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi })).$(8)

The computation of first multiple Zagreb $(\mathit{P}{M}_1)$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{P}{M}_1(\mathit{Cu}{F}_2)=\prod _{\mathit{i}=1}^3\prod _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi }))$

$=(8)\times (\mathit{m}+\mathit{n}+1)\times (20)\times (\mathit{m}+\mathit{n}-2)\times (36)$

$\times (2\mathit{mn}-\mathit{m}-\mathit{n}+1)=120(2\mathit{m}+2\mathit{n}+2)(4\mathit{m}+4\mathit{n}-8)\text{break}(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6).$

The computation of second multiple Zagreb $(\mathit{P}{M}_2)$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{P}{M}_2(\mathit{Cu}{F}_2)=\prod _{\mathit{i}=1}^3\prod _{\varphi \psi \in E_i(\mathit{G})}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }))$

$=(6)\times (\mathit{m}+\mathit{n}+1)\times (24)\times (\mathit{m}+\mathit{n}-2)\times (54\times (2\mathit{mn}-\mathit{m}-\mathit{n}+1)$

$=162(2\mathit{m}+2\mathit{n}+2)(4\mathit{m}+4\mathit{n}-8)(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6).$

Table 4 and Figure 6 present the numerical analysis and graphical depiction of the hyper-Zagreb, $\mathit{P}{M}_1$, and $\mathit{P}{M}_2$, respectively.

Figure 6. Graphical depiction of $\mathit{HM}(\mathit{Cu}{F}_2)$, $\mathit{P}{M}_1(\mathit{Cu}{F}_2)$, and $\mathit{P}{M}_2(\mathit{Cu}{F}_2)$.

Table 4. Numerical comparative analysis of $\mathit{HM}(\mathit{Cu}{F}_2)$, $\mathit{P}{M}_1(\mathit{Cu}{F}_2),$ and $\mathit{P}{M}_2(\mathit{Cu}{F}_2)$.

$[\mathit{m},\mathit{n}]$	$\mathit{HM}(\mathit{Cu}{F}_2)$	$\mathit{P}{M}_1(\mathit{Cu}{F}_2)$	$\mathit{P}{M}_2(\mathit{Cu}{F}_2)$
(Ahmad et al. 2021, 2021)	312	0	0
(Zaheer et al. 2020, 2020)	1440	288,000	388,800
(Ali Malik et al. 2023, 2023)	3432	2,096,640	2,830,464
(Ami et al. 1998, 1998)	6288	7,776,000	10,497,600
(Balaban 1982, 1982)	10,008	20,782,080	28,055,808
(Balaban and Quintas 1983, 1983)	14,592	45,676,800	61,663,680
(Bollobs and Erds 1998, 1998)	20,040	88,128,000	118,972,800
(Das and Gutman 2004, 2004)	26,352	154,909,440	209,127,744

The forgotten index is computed by Furtula and Gutman (2015) as follows:

(9) $\mathit{F}(\mathit{G})=\sum _{\varphi \psi \in E(Cu{F}_2)}(\mathcal{\S }(\mathit{\varphi }{)}^2+\mathcal{\S }(\mathit{\psi }{)}^2).$(9)

The computation of forgotten $(\mathit{F})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{F}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}(\mathcal{\S }(\mathit{\varphi }{)}^2+\mathcal{\S }(\mathit{\psi }{)}^2)$

$=(1^2+3^2)(2\mathit{m}+2\mathit{n}+2)+(2^2+3^2)(4\mathit{m}+4\mathit{n}-8)$

$+(3^2+3^2)(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)=216\mathit{mn}-36\mathit{m}-36\mathit{n}+24.$

(Wang, Huang, and Liu 2012)Vukičević et al. (2009) defined the augmented Zagreb index as follows:

(10) $\mathit{AZI}(\mathit{G})=\sum _{\varphi \psi \in E(Cu{F}_2)}{\left[\frac{(\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi ))}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )-2}\right]}^3.$(10)

The computation of augmented Zagreb $(\mathit{AZI})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{AZI}(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}{\left[\frac{(\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi ))}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )-2}\right]}^3$

$={\left(\frac{1\times 3}{1+3-2}\right)}^3(2\mathit{m}+2\mathit{n}+2)+(2{)}^3(4\mathit{m}+4\mathit{n}-8)+{\left(\frac{9}{7}\right)}^3(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=\frac{2187\mathit{mn}}{16}-\frac{947\mathit{m}}{32}-\frac{947\mathit{n}}{32}+\frac{355}{32}.$

The Balaban index (Balaban 1982; Balaban and Quintas 1983) was developed by Balaban as follows:

(11) $\mathit{J}(\mathit{G})=\left(\frac{\mathit{y}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{y}{ }^{\mathrm{\prime }}-\mathit{x}{ }^{\mathrm{\prime }}+2}\right)\left[\sum _{\varphi \psi \in E(Cu{F}_2)}\frac{1}{\sqrt{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}\right].$(11)

The computation of Balaban $(\mathit{J})$ index for $\mathit{Cu}{F}_2$ is as follows:

$\mathit{J}(\mathit{Zr}{O}_2)=\left(\frac{\mathit{y}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{y}{ }^{\mathrm{\prime }}-\mathit{x}{ }^{\mathrm{\prime }}+2}\right)\left[\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Zr}{O}_2)}\frac{1}{\sqrt{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}}\right]$

$=\left(\frac{12\mathit{mn}}{4\mathit{mn}-2\mathit{m}-2\mathit{n}+2}\right)\frac{1}{\sqrt{(1\times 3)}}\times (2\mathit{m}+2\mathit{n}+2)+\frac{1}{\sqrt{(2\times 3)}}$

$\times (4\mathit{m}+4\mathit{n}-8)+\frac{1}{\sqrt{(3\times 3)}}\times (12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=\left(\frac{48m^2{n}^2+9.452324m^2\mathit{n}+9.452324\mathit{m}{n}^2-1.335429\mathit{mn}}{4\mathit{mn}-2\mathit{m}-2\mathit{n}+2}\right).$

Table 5 compares the forgotten $(\mathit{F})$, augmented Zagreb $(\mathit{AZI})$, and Balaban $(\mathit{J})$ indices numerically, while Figure 7 compares them graphically.

Figure 7. Graphical depiction of $\mathit{F}(\mathit{Cu}{F}_2)$, $\mathit{AZI}(\mathit{Cu}{F}_2)$, and $\mathit{J}(\mathit{Cu}{F}_2)$.

Table 5. Numerical comparative analysis of $\mathit{F}(\mathit{Cu}{F}_2)$, $\mathit{AZI}(\mathit{Cu}{F}_2)$, and $\mathit{J}(\mathit{Cu}{F}_2)$.

$[\mathit{m},\mathit{n}]$	$\mathit{F}(\mathit{Cu}{F}_2)$	$\mathit{AZI}(\mathit{Cu}{F}_2)$	$\mathit{J}(\mathit{Cu}{F}_2)$
(Ahmad et al. 2021, 2021)	168	88.59375	32.78461
(Zaheer et al. 2020, 2020)	744	439.46875	91.389547
(Ali Malik et al. 2023, 2023)	1752	1063.71875	168.707948
(Ami et al. 1998, 1998)	3192	1961.34375	269.530613
(Balaban 1982, 1982)	5664	3132.34375	394.264578
(Balaban and Quintas 1983, 1983)	7368	4576.71875	542.978104
(Bollobs and Erds 1998, 1998)	10,104	6294.46875	715.687403
(Das and Gutman 2004, 2004)	13,272	8285.59375	912.396959

First, second, and third redefined Zagreb type indices were presented by Ranjini, Lokesha, and Usha (2013) as follows:

(12) $\mathit{ReZ}{G}_1(\mathit{G})=\sum _{\varphi \psi \in E(Cu{F}_2)}\frac{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}.$(12)

(13) $\mathit{ReZ}{G}_2(\mathit{G})=\sum _{\varphi \psi \in E(Cu{F}_2)}\frac{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}.$(13)

(14) $\mathit{ReZ}{G}_3(\mathit{G})=\sum _{\varphi \psi \in E(Cu{F}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }))(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi })).$(14)

The computation of redefined first Zagreb $(\mathit{ReZ}{G}_1)$ index is as follows:

$\mathit{ReZ}{G}_1(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}\frac{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}$

$=\left(\frac{8}{1\times 3}\right)(\mathit{m}+\mathit{n}+1)+\left(\frac{20}{2\times 3}\right)(\mathit{m}+\mathit{n}-2)+\left(\frac{36}{3\times 3}\right)(2\mathit{mn}-\mathit{m}-\mathit{n}+1)$

$=8\mathit{mn}+2\mathit{m}+2\mathit{n}.$

The computation of redefined second Zagreb $(\mathit{ReZ}{G}_2)$ index is as follows:

$\mathit{ReZ}{G}_2(\mathit{Cu}{F}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Cu}{F}_2)}\frac{\mathcal{\S }(\varphi )\times \mathcal{\S }(\psi )}{\mathcal{\S }(\varphi )+\mathcal{\S }(\psi )}$

$=\left(\frac{3}{8}\right)(\mathit{m}+\mathit{n}+1)+\left(\frac{24}{5}\right)(\mathit{m}+\mathit{n}-2)+\left(\frac{54}{6}\right)$

$(2\mathit{mn}-\mathit{m}-\mathit{n}+1)=18\mathit{mn}-\frac{27\mathit{m}}{10}-\frac{27\mathit{n}}{10}+\frac{9}{10}.$

The computation of redefined third Zagreb $(\mathit{ReZ}{G}_3)$ index is as follows:

$\mathit{ReZ}{G}_3(\mathit{Zr}{O}_2)=\sum _{\mathit{i}=1}^3\sum _{\varphi \psi \in E_i(\mathit{Zr}{O}_2)}(\mathcal{\S }(\mathit{\varphi })\times \mathcal{\S }(\mathit{\psi }))(\mathcal{\S }(\mathit{\varphi })+\mathcal{\S }(\mathit{\psi }))$

$=(1\times 3)(1+3)(2\mathit{m}+2\mathit{n}+2)+(2\times 3)(2+3)(4\mathit{m}+4\mathit{n}-8)$

$+(3\times 3)(3+3)(12\mathit{mn}-6\mathit{m}-6\mathit{n}+6)$

$=648\mathit{mn}-180\mathit{m}-180\mathit{n}+108.$

Table 6 and Figure 8 provide the numerical comparison and graphical representation of $\mathit{ReZ}{G}_1$, $\mathit{ReZ}{G}_2$, and $\mathit{ReZ}{G}_3$ indices, respectively.

Figure 8. Graphical depiction of $\mathit{ReZ}{G}_2(\mathit{Cu}{F}_2)$ and $\mathit{ReZ}{G}_3(\mathit{Cu}{F}_2)$.

Table 6. Numerical comparative analysis of $\mathit{ReZ}{G}_1(\mathit{Cu}{F}_2)$, $\mathit{ReZ}{G}_2(\mathit{Cu}{F}_2)$, and $\mathit{ReZ}{G}_3(\mathit{Cu}{F}_2)$.

$[\mathit{m},\mathit{n}]$	$\mathit{ReZ}{G}_1(\mathit{Cu}{F}_2)$	$\mathit{ReZ}{G}_2(\mathit{Cu}{F}_2)$	$\mathit{ReZ}{G}_3(\mathit{Cu}{F}_2)$
(Ahmad et al. 2021, 2021)	12	13.5	396
(Zaheer et al. 2020, 2020)	40	62.1	1980
(Ali Malik et al. 2023, 2023)	84	146.7	4860
(Ami et al. 1998, 1998)	144	267.3	9036
(Balaban 1982, 1982)	220	423.9	14,508
(Balaban and Quintas 1983, 1983)	312	616.5	21,276
(Bollobs and Erds 1998, 1998)	420	845.1	29,340
(Das and Gutman 2004, 2004)	544	1109.7	38,700

Analyzing the Rational Curve Fitting $\left({\mathbf{R}}_{\mathit{cf}}\right)$ Relationships Between Indices and Heat of Formation $\left(\mathrm{\Delta }{H}_f\right)$

Computational chemistry researchers have utilized rational curve fitting techniques to gain a deeper understanding of the connection between molecular descriptors and the thermodynamic parameters that go along with them. One such application is establishing a connection between the heat of chemical compound synthesis and topological indices such as the Randić or Zagreb indices. Using methodical analysis and data-driven approaches, researchers have tried to find patterns and trends that link the thermodynamic stability of molecules, which is demonstrated by the heat of formation to their structural properties, as reflected by these indices.

Researchers seek to develop mathematical models that accurately capture the intricate relationships between molecular topology and energetics by applying methods of rational curve fitting. These studies contribute to our understanding of molecular dynamics and may have consequences for the rational design of novel compounds with desired thermodynamic features. A number of significant factors are necessary for assessing the quality and dependability of the derived mathematical models. One common metric used to quantify how much of the variance in the dependent variable (heat of formation) can be anticipated from the independent variable (topological indices) is the coefficient of determination $(R^2)$. A higher $R^2$ value indicates a better fit of the curve to the data, indicating the effectiveness of the chosen indices in explaining the variations in heat of formation. The root mean square error, or $(\mathit{RMSE})$, is a widely used statistic that evaluates the average divergence between the expected values from the curve and the actual observed values. A lower $\mathit{RMSE}$ indicates a more accurate and exact model.

The sum of squares $(\mathit{SSE})$ quantifies the total difference between the values observed and the values predicted by the regression model. It is computed as the sum of the squared differences between the actual and predicted values. A lower SSE indicates a closer fit between the model and the data. The modified $R^2$ takes the number of predictors in the model into account. It penalizes the addition of unneeded variables that do not significantly improve the variance’s explanation. According to standard circumstances, the standard enthalpy of gas for $\mathit{Cu}{F}_2$ is $-266.94\mathit{KJ}/\mathit{mol}$ (Jenkins and Donald 2005). Below are the models that were explained regarding the relation between indices vs. $\mathrm{\Delta }{H}_f$ with the help of MATLAB. Figures 9–17 show the graphical behavior. Let the standardized mean be represented with the symbol $\mathit{\eta }$ $(\mathit{eta})$ and the standard deviation with $\mathit{\rho }$ $(\mathit{rho})$.

Figure 9. (a) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $R_1(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $R_{-}1(\mathit{Cu}{F}_2)$.

Figure 10. (a) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $R_{\frac{1}{2}}(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $R_{\frac{-1}{2}}(\mathit{Cu}{F}_2)$.

Figure 11. (a) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $M_1(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_{\mathit{cf}}$ between $\mathrm{\Delta }{H}_f$ and $M_2(\mathit{Cu}{F}_2)$.

Figure 12. (a) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{ABC}(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{GA}(\mathit{Cu}{F}_2)$.

Figure 13. ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{P}{M}_1(\mathit{Cu}{F}_2)$.

Figure 14. (a) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{P}{M}_2(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{HM}(\mathit{Cu}{F}_2)$.

Figure 15. (a) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ of $\mathit{F}(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ of $\mathit{J}(\mathit{Cu}{F}_2)$.

Figure 16. (a) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{AZI}(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{ReZ}{G}_1(\mathit{Cu}{F}_2)$.

Figure 17. (a) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{ReZ}{G}_2(\mathit{Cu}{F}_2)$ and (b) ${\mathbf{R}}_c\mathit{f}$ between $\mathrm{\Delta }{H}_f$ and $\mathit{ReZ}{G}_3(\mathit{Cu}{F}_2)$.

$\mathit{f}(R_1)=\frac{({\beta }_1\times R_1^4+{\beta }_2\times R_1^3+{\beta }_3\times R_1^2+{\beta }_4\times R_1+{\beta }_5)}{(R_1+{\gamma }_1)}$

where $R_1$ is standardized by $\mathit{\eta }=2550$ and $\mathit{\rho }=2324$. The coefficients: ${\beta }_1=-0.03033(-0.07318,0.01251)$, ${\beta }_2=0.07434(0.03215,0.1165)$, ${\beta }_3=-3.964(-4.043,-3.884)$, ${\beta }_4=-6.354(-6.514,-6.194)$, ${\beta }_5=-2.023(-2.159,-1.888)$, ${\gamma }_1=0.4433(0.4113,\mathit{o}.4754)$.

$\mathit{SSE}:0.000319$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01263$

$\mathit{f}(R_{-1})=\frac{({\beta }_1\times R_{-1}^4+{\beta }_2\times R_{-1}^3+{\beta }_3\times R_{-1}^2+{\beta }_4\times R_{-1}+{\beta }_5)}{(R_{-1}+{\gamma }_1)}$

where $R_{-1}$ is standardized by $\mathit{\eta }=40$ and $\mathit{\rho }=33.31$. The coefficients: ${\beta }_1=0.04993(-0.007057,0.1069)$, ${\beta }_2=-0.1337(-0.1907,-0.07664)$, ${\beta }_3=-4.063(-4.171,-3.955)$, ${\beta }_4=-5.937(-6.172,-5.702)$, ${\beta }_5=-1.694(-1.886,-1.502)$, ${\gamma }_1=0.3823(0.3354,\mathit{o}.4293)$.

$\mathit{SSE}:0.0006083$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01744$.

$\mathit{f}(R_{\frac{1}{2}})=\frac{({\beta }_1\times R_{\frac{1}{2}}^3+{\beta }_2\times R_{\frac{1}{2}}^2+{\beta }_3\times R_{\frac{1}{2}}+{\beta }_4)}{(R_{\frac{1}{2}}^2+{\gamma }_1\times R_{\frac{1}{2}}+{\gamma }_2)}$

In which $R_{\frac{1}{2}}$ is standardized by $\mathit{\eta }=877.2$ and $\mathit{\rho }=790.4$. The coefficients: ${\beta }_1=-3.998(-4.116,-3.881)$, ${\beta }_2=-3.844(-18.76,11.07)$, ${\beta }_3=1.726(-21.52,24.97)$, ${\beta }_4=1.103(-6.076,8.281)$, ${\gamma }_1=-0.1674(-3.896,3.561)$, ${\gamma }_2=-0.2434(-1.824,1.338)$.

$\mathit{SSE}:0.002717$, $R^2:1$, $\mathit{Adjusted}{R}^2:0.9999$, and $\mathit{RMSE}:0.03686$.

$\mathit{f}(R_{\frac{-1}{2}})=\frac{({\beta }_1\times R_{\frac{-1}{2}}^3+{\beta }_2\times R_{\frac{-1}{2}}^2+{\beta }_3\times R_{\frac{-1}{2}}+{\beta }_4)}{(R_{\frac{-1}{2}}+{\gamma }_1)}$

In which $R_{\frac{-1}{2}}$ is standardized by $\mathit{\eta }=109$ and $\mathit{\rho }=94.11$. The coefficients: ${\beta }_1=-0.04467(-0.08055,-0.008784)$, ${\beta }_2=-3.992(-4.024,-3.96)$,${\beta }_3=-6.091(-6.256,-5.926)$, ${\beta }_4=-1.818(-1.955,-1.682)$, ${\gamma }_1=0.4047(0.3717,0.4378)$.

$\mathit{SSE}:0.0009139$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01745$.

Regression analysis techniques are used to find the coefficients of the model after a dataset including known heat of formation values for various molecules is collected and their Randić indices are computed. It is noteworthy that the particular manifestation of the correlation may differ based on the dataset and the underlying chemistry under investigation. A number of measures, including the adjusted $R^2$ $(\mathit{adj}{R}^2)$, root mean square error $(\mathit{RMSE})$, sum of squares error $(\mathit{SSE})$, and coefficient of determination $(R^2)$, can be used to evaluate the model’s accuracy and dependability. By looking at the Randić index, these measurements show how effectively the model accounts for the variance in the heat of formation.

$\mathit{f}(M_1)=\frac{({\beta }_1\times M_1^5+{\beta }_2\times M_1^4+{\beta }_3\times M_1^3+{\beta }_4\times M_1^2+{\beta }_5\times M_1+{\beta }_6)}{(M_1+{\gamma }_1)}$

where $M_1$ is standardized by $\mathit{\eta }=1768$ and $\mathit{\rho }=1588$.

The coefficients: ${\beta }_1=0.01035(-0.1106,0.1313)$, ${\beta }_2=-0.02314(-0.1391,0.09286)$, ${\beta }_3=0.01272(-0.2631,0.2886)$, ${\beta }_4=-3.976(-4.111,-3.842)$, ${\beta }_5=-6.247(-6.635,-5.859)$, ${\beta }_6=-1.94(-2.223,-1.658)$, ${\gamma }_1=0.4273(0.3622,0.4923)$.

$\mathit{SSE}:3.106\times {10}^{-6}$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.005574$.

$\mathit{f}(M_2)=\frac{({\beta }_1\times M_2^4+{\beta }_2\times M_2^3+{\beta }_3\times M_2^2+{\beta }_4\times M_2+{\beta }_5)}{(M_2+{\gamma }_1)}$

where $M_2$ is standardized by $\mathit{\eta }=2550$ and $\mathit{\rho }=2324$. The coefficients: ${\beta }_1=-0.03033(-0.07318,0.01251)$, ${\beta }_2=0.07434(0.03215,0.1165)$, ${\beta }_3=-3.964(-4.043,-3.884)$, ${\beta }_4=-6.354(-6.514,-6.194)$, ${\beta }_5=-2.023(-2.159,-1.888)$, ${\gamma }_1=0.4433(0.4113,0.4754)$.

$\mathit{SSE}:0.000319$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01263$.

Chemical process optimization requires a thorough understanding of the heat of formation. Chemical process optimization can be aided by the employment of models based on Zagreb-type indices to forecast the thermodynamic characteristics of reaction intermediates. The evaluation of the environmental impact of different chemical compounds can be aided by predictive models based on Zagreb-type criteria.

$\mathit{f}(\mathit{ABC})=\frac{({\beta }_1\times \mathit{AB}{C}^2+{\beta }_2\times \mathit{ABC}+{\beta }_3)}{(\mathit{AB}{C}^4+{\gamma }_1\times \mathit{AB}{C}^3+{\gamma }_2\times \mathit{AB}{C}^2+{\gamma }_3\times \mathit{ABC}+{\gamma }_4)}$

In which $\mathit{ABC}$ is standardized by $\mathit{\eta }=208.1$ and $\mathit{\rho }=182.9$.

The coefficients: ${\beta }_1=-1.065\mathit{e}+05(-8.545\mathit{e}+07,8.524\mathit{e}+07)$, ${\beta }_2=-1.65\mathit{e}+05(-1.325\mathit{e}+08,1.322\mathit{e}+08)$, ${\beta }_3=-5.032\mathit{e}+04(-4.04\mathit{e}+07,4.03\mathit{e}+07)$, ${\gamma }_1=42.26(-3.54\mathit{e}+04,3.549\mathit{e}+04)$, ${\gamma }_2=-151.5(-1.216\mathit{e}+05,1.213\mathit{e}+05)$, ${\gamma }_3=2.671\mathit{e}+04(-2.139\mathit{e}+07,2.144\mathit{e}+07)$, ${\gamma }_4=1.116\mathit{e}+04(-8.94\mathit{e}+06,8.963\mathit{e}+06)$.

$\mathit{SSE}:3.073\times {10}^{-5}$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.005543$.

The potential correlation or relationship between the atomic bomb connectivity index and the heat of production can be investigated by using a curve fitting model. A mathematical representation of the relationship between any underlying patterns or trends in the data will be provided by the model.

$\mathit{f}(\mathit{GA})=\frac{({\beta }_1\times \mathit{G}{A}^2+{\beta }_2\times \mathit{GA}+{\beta }_3)}{(\mathit{GA}+{\gamma }_1)}$

where $\mathit{GA}$ is standardized by $\mathit{\eta }=302.8$ and $\mathit{\rho }=269.3$.

The coefficients: ${\beta }_1=-4(-4.009,3.992)$, ${\beta }_2=-6.195(-6.228,-6.162)$, ${\beta }_3=-1.896(-1.927,-1.865)$, ${\gamma }_1=0.4192(0.4113,0.4271)$.

$\mathit{SSE}:0.0001463$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.006047$.

When combined with machine learning techniques or regression models, the GA index can be used as a predictive tool to estimate the heat of production of novel or untested compounds. Predicting thermodynamic stability is critical in drug development and materials science, where this is especially useful.

$\mathit{f}(\mathit{P}{M}_1)=\frac{({\beta }_1\times \mathit{P}{M}_1^3+{\beta }_2\times \mathit{P}{M}_1^2+{\beta }_3\times \mathit{P}{M}_1+{\beta }_4)}{(\mathit{P}{M}_1^3+{\gamma }_1\times \mathit{P}{M}_1^2+{\gamma }_2\times \mathit{P}{M}_1+{\gamma }_3)}$

In which $\mathit{P}{M}_1$ is standardized by $\mathit{\eta }=3.996\times {10}^7$ and $\mathit{\rho }=5.552\times {10}^7$. The coefficients: ${\beta }_1=-20.59(-43.81,2.638)$, ${\beta }_2=-45.85(-132.7,40.99)$, ${\beta }_3=-32.81(-122.2,56.54)$, ${\beta }_4=-7.539(-35.1,20.02)$, ${\gamma }_1=4.746(-3.68,13.17)$, ${\gamma }_2=4.688(-7.691,17.07)$, ${\gamma }_3=1.28(-3.347,5.907)$.

$\mathit{SSE}:0.01744$, $R^2:0.9998$, $\mathit{Adjusted}{R}^2:0.9989$, and $\mathit{RMSE}:0.1321$.

$\mathit{f}(\mathit{P}{M}_2)=\frac{({\beta }_1\times \mathit{P}{M}_2^2+{\beta }_2\times \mathit{P}{M}_2+{\beta }_3)}{(\mathit{P}{M}_2^2+{\gamma }_1\times \mathit{P}{M}_2+{\gamma }_2)}$

where $\mathit{P}{M}_2$ is standardized by $\mathit{\eta }=5.394\times {10}^7$ and $\mathit{\rho }=7.495\times {10}^7$. The coefficients: ${\beta }_1=-18.18(-24.17,-12.2)$, ${\beta }_2=-29.35(-42.84,-15.86)$, ${\beta }_3=-11.71(-18.44,-4.972)$, ${\gamma }_1=3.44(1.759,5.122)$,${\gamma }_2=1.977(0.7087,3.245)$.

$\mathit{SSE}:0.2148$, $R^2:0.9981$, $\mathit{Adjusted}{R}^2:0.9956$, and $\mathit{RMSE}:0.2676$.

Computational models can be improved and validated by utilizing the relationship between heat of formation and many Zagreb-type variables. By contrasting quantum chemical computations with empirical connections acquired from actual data, it helps to measure the correctness of these calculations. Property prediction in materials science: It can be used to forecast and create materials with desired thermodynamic properties when the materials have certain thermal properties.

$\mathit{f}(\mathit{HM})=\frac{({\beta }_1\times \mathit{H}{M}^4+{\beta }_2\times \mathit{H}{M}^3+{\beta }_3\times \mathit{H}{M}^2+{\beta }_4\times \mathit{HM}+{\beta }_5)}{(\mathit{HM}+{\gamma }_1)}$

In which $\mathit{HM}$ is standardized by $\mathit{\eta }=1.031\times {10}^4$ and $\mathit{\rho }=9355$. The coefficients: ${\beta }_1=-0.02621(-0.06287,0.01045)$, ${\beta }_2=0.06445(0.02829,0.1006)$, ${\beta }_3=-3.969(-4.037,-3.901)$, ${\beta }_4=-6.333(-6.471,-6.196)$, ${\beta }_5=-2.007(-2.123,-1.89)$, ${\gamma }_1=0.4403(0.4127,0.4678)$.

$\mathit{SSE}:0.0002344$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01083$.

In chemoinformatics, the correlation between $\mathit{HM}$ and heat of formation might be useful for analyzing vast chemical databases. It makes it possible to identify structural patterns linked to specific thermodynamic features, which makes it easier to find new compounds with desirable attributes.

$\mathit{f}(\mathit{F})=\frac{({\beta }_1\times F^3+{\beta }_2\times F^2+{\beta }_3\times \mathit{F}+{\beta }_4)}{(F^2+{\gamma }_1\times \mathit{F}+{\gamma }_2)}$

where $\mathit{F}$ is standardized by $\mathit{\eta }=5283$ and $\mathit{\rho }=4708$. The coefficients: ${\beta }_1=-4.015(-4.076,-3.953)$, ${\beta }_2=-5.943(-6.259,-5.627)$, ${\beta }_3=-1.284(-1.744,-0.8244)$, ${\beta }_4=0.3306(0.06361,0.5975)$, ${\gamma }_1=0.3505(0.2742,0.4268)$, ${\gamma }_2=-0.07681(-0.1424,-0.01125)$.

$\mathit{SSE}:0.001133$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.0238$.

One important thermodynamic factor that affects a molecule’s chemical reactivity is the heat of formation. One may be able to predict the stability or reactivity of chemical compounds by developing a relationship with the forgotten index.

$\mathit{f}(\mathit{J})=\frac{({\beta }_1\times J^5+{\beta }_2\times J^4+{\beta }_3\times J^3+{\beta }_4\times J^2+{\beta }_5\times \mathit{J}+{\beta }_6)}{(\mathit{J}+{\gamma }_1)}$

where $\mathit{J}$ is standardized by $\mathit{\eta }=391$ and $\mathit{\rho }=312.3$.

The coefficients: ${\beta }_1=-0.06849(-0.7771,0.6402)$, ${\beta }_2=0.1332(-0.4538,0.7202)$, ${\beta }_3=-0.04784(-1.781,1.686)$, ${\beta }_4=-4.156(-4.988,-3.324)$, ${\beta }_5=-5.978(-8.628,-3.328)$, ${\beta }_6=-1.702(-3.608,0.2034)$, ${\gamma }_1=0.3887(-0.06927,0.8467)$.

$\mathit{SSE}:0.001192$, $R^2:1$, $\mathit{Adjusted}{R}^2:0.9999$, and $\mathit{RMSE}:0.03452$.

Examine the model’s coefficients and the importance of the Balaban index. This may shed light on the ways in which the Balaban index captures topological features that affect molecules’ thermodynamic stability. Based on the compounds’ Balaban indices, use the created model to forecast the heat of production for new ones. This predictive power can be useful in areas such as materials science, medication creation, and other areas where knowledge of thermodynamic stability is essential.

$\mathit{f}(\mathit{AZI})=\frac{({\beta }_1\times \mathit{AZ}{I}^4+{\beta }_2\times \mathit{AZ}{I}^3+{\beta }_3\times \mathit{AZ}{I}^2+{\beta }_4\times \mathit{AZI}+{\beta }_5)}{(\mathit{AZI}+{\gamma }_1)}$

where $\mathit{AZI}$ is standardized by $\mathit{\eta }=3230$ and $\mathit{\rho }=2945$.

The coefficients: ${\beta }_1=-0.02948(-0.07103,0.01208)$, ${\beta }_2=0.07229(0.03136,0.1132)$, ${\beta }_3=-3.965(-4.042,-3.888)$, ${\beta }_4=-6.35(-6.505,-6.194)$, ${\beta }_5=-2.02(-2.151,-1.889)$, ${\gamma }_1=0.4427(0.4116,0.4738)$.

$\mathit{SSE}:0.0003003$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01225$.

The correlation between heat of formation and the augmented Zagreb index can help forecast the stability and energetics of drug candidates, which is important in the drug discovery process where knowledge of molecular characteristics is essential. Selecting prospective medication candidates with favorable thermodynamic characteristics requires an understanding of this.

$\mathit{f}(\mathit{ReZ}{G}_1)=\frac{({\beta }_1\times \mathit{ReZ}{G}_1^3+{\beta }_2\times \mathit{ReZ}{G}_1^2+{\beta }_3\times \mathit{ReZ}{G}_1+{\beta }_4)}{(\mathit{ReZ}{G}_1^2+{\gamma }_1\times \mathit{ReZ}{G}_1+{\gamma }_2)}$

where $\mathit{ReZ}{G}_1$ is standardized by $\mathit{\eta }=222$ and $\mathit{\rho }=190.2$.

The coefficients: ${\beta }_1=-4.011(-4.187,-3.834)$, ${\beta }_2=-8.768(-16.86,-0.6751)$, ${\beta }_3=-5.886(-18.09,6.313)$, ${\beta }_4=-1.258(-4.84,2.323)$, ${\gamma }_1=1.053(-0.9747,3.08)$, ${\gamma }_2=0.2821(-0.5183,1.082)$.

$\mathit{SSE}:0.003189$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.03993$.

$\mathit{f}(\mathit{ReZ}{G}_2)=\frac{({\beta }_1\times \mathit{ReZ}{G}_2^4+{\beta }_2\times \mathit{ReZ}{G}_2^3+{\beta }_3\times \mathit{ReZ}{G}_2^2+{\beta }_4\times \mathit{ReZ}{G}_2+{\beta }_5)}{(\mathit{ReZ}{G}_2+{\gamma }_1)}$

In which $\mathit{ReZ}{G}_2$ is standardized by $\mathit{\eta }=435.6$ and $\mathit{\rho }=393.6$. The coefficients: ${\beta }_1=-0.01982(-0.04712,0.007477)$, ${\beta }_2=0.04899(0.02201,0.07598)$, ${\beta }_3=-3.977(-4.028,-3.926)$, ${\beta }_4=-6.302(-6.405,-6.198)$, $\mathit{\beta }=-1.981(-2.068,-1.894)$, ${\gamma }_1=0.4355(0.4148,0.4562)$.

$\mathit{SSE}:0.0001307$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.008084$.

$\mathit{f}(\mathit{ReZ}{G}_3)=\frac{({\beta }_1\times \mathit{ReZ}{G}_3^4+{\beta }_2\times \mathit{ReZ}{G}_3^3+{\beta }_3\times \mathit{ReZ}{G}_3^2+{\beta }_4\times \mathit{ReZ}{G}_3+{\beta }_5)}{(\mathit{ReZ}{G}_3+{\gamma }_1)}$

In which $\mathit{ReZ}{G}_3$ is standardized by $\mathit{\eta }=1.501\times {10}^4$ and $\mathit{\rho }=1.377\times {10}^4$. The coefficients: ${\beta }_1=-0.03888(-0.09493,0.01716)$, ${\beta }_2=0.09467(0.03966,0.1947)$, ${\beta }_3=-3.953(-4.057,-3.849)$, ${\beta }_4=-6.396(-6.602,-6.189)$, ${\beta }_5=-2.058(-2.233,-1.882)$ ${\gamma }_1=0.4497(0.4082,0.4911)$.

$\mathit{SSE}:0.0005417$, $R^2:1$, $\mathit{Adjusted}{R}^2:1$, and $\mathit{RMSE}:0.01646$.

In examining a redefined Zagreb-type index model, scientists are essentially investigating how particular adjustments or redefinitions of the Zagreb indices can improve the model’s ability to anticipate formation heat changes. Redefining the indices could involve adding more structural details or using different topological descriptors to better represent the molecular characteristics affecting thermodynamic stability.

Conclusion

We utilized edge partitioning to generate indices for $\mathit{Cu}{F}_2$, presenting both the heat of formation (HOF) and closed formulas alongside degree-based topological indices. These estimations were subjected to comparative assessments using MATLAB for numerical computations and Maple for graphical representations. We used curve fitting techniques to create relationships between different indexes and the volatility levels that corresponded with them. Performance was assessed using several mean square error techniques, and the findings showed that the logical method consistently produced better outcomes, even with differences in parameterization settings. Notably, when comparing mobility to indices, the rational fit method turned out to be the most accurate. This thorough understanding of how the geometry of $\mathit{Cu}{F}_2$ influences its properties provides a helpful mathematical tool to facilitate more efficient structural alterations in a range of applications.

Authors Contributions

All authors made equal contributions to this work.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.