Investigating Seidel energies and thermodynamic properties of benzenoid hydrocarbons through regression models

Abstract

In this article, we present the concept of extended Seidel energy by employing a generalized extended matrix to study various molecular properties, including the Kovats retention index, boiling point, enthalpy of formation, entropy, acentric factor, and octanol-water partition coefficient. Our research broadens the scope of energy matrices in graph theory, with a particular emphasis on Sombor energy, reduced Sombor energy, average Sombor energy, Banhatti Sombor energy and reduced Banhatti Sombor energy. We examined the correlation of these graph-based energy descriptors with the thermodynamic properties of Benzenoid hydrocarbons (BHC), uncovering strong relationships between these indices and different molecular attributes. These results provide valuable insights into the predictive potential of graph energies in modeling the thermodynamic behavior of polycyclic aromatic hydrocarbons.

Introduction

Graph theory, a branch of mathematics, focuses on studying the properties and relationships between objects, typically represented as vertices (or nodes) and the connections between them, known as edges. This field has become a crucial tool in various disciplines, including computer science, biology, social science, and engineering. A graph is composed of two fundamental elements: a set of vertices and a set of edges, with each edge connecting two vertices. Graphs can be either directed, where connections have a specific direction, or undirected, where connections are bidirectional. Graph theory is widely applied to solve practical problems such as network design, shortest path calculations, social network analysis, and the modeling of biological systems¹. By offering a visual and mathematical framework to explore relationships, graph theory continues to drive advancements in research and technology.

Chemical graph theory is an interdisciplinary field that merges concepts from chemistry and graph theory to study Molecular structures. In this field, molecules are modeled as graphs, where atoms serve as vertices and chemical bonds as edges. Graphs can be classified as directed, where connections have a specific direction, or undirected, where connections are bidirectional. In chemical graph theory, we focus on undirected graphs, as they represent molecular structures in which the bonds between atoms do not have a direction. This representation simplifies complex Molecular structures, enabling researchers to use mathematical tools to analyze and predict chemical properties, behaviors, and reactions. Chemical graph theory plays a pivotal role in modern chemistry, serving as the foundation for the development of algorithms and computational tools². This approach transforms complex molecular compositions into abstract mathematical representations, enabling researchers to manipulate and analyze molecular architectures through graph theory techniques³. As a result, it provides deep insights into various chemical phenomena, including molecular properties, reaction mechanisms, and the relationship between structure and function⁴. This methodology has revolutionized the study of chemistry, driving significant advancements in material design, drug discovery, and the understanding of fundamental chemical concepts.

Recent innovation in catalysis research has been focused on environmental remediation, material science, and synthetic chemistry via Seidel energies. Both Zhou et al.⁵ and Hua et al.⁶ deal with catalytic oxidation: the former presents a solution to mitigate the poisoning problem by chlorine in the degradation of VOCs, while the latter deals with the challenge of dual removal of chlorobenzene and ammonia in SCR exhaust gases and provides a pathway to more efficient pollution control via Seidel energies. Meanwhile, Liu et al.⁷ describe the donor-acceptor hybrid materials based on naphthalenediimide and perylenediimide, focusing on their potential in optoelectronics and energy storage concerning their versatility in structure and electronics. Zhao et al.⁸ further contribute a novel aminoselenation method to cyclohexanones, offering a sustainable pathway to aromatic compounds via Seidel energies.

Topological descriptors, derived from the connectivity of nodes within molecules, are crucial tools for simplifying complex chemical structures with predictive accuracy. These indices play an essential role in quantitative structure-activity relationships (QSAR) by concisely encoding structural information, thus offering insights into various physicochemical properties without the need for extensive experimental data. Researchers utilize topological indices^9,10 to explore molecular characteristics such as boiling points, solubility, toxicity, and biological activity. This approach supports the design of new molecules with desired properties, benefiting fields such as environmental science, materials science, and medicinal chemistry. As fundamental elements of computational chemistry, topological indices are indispensable for understanding and predicting molecular behavior across diverse chemical contexts. Most commonly, node-degree-based topological descriptors are expressed as:

$$\begin{aligned} TI\left( G \right) =\sum \limits _{{{\mu }_{i}}{{\mu }_{j}}\in E\left( G \right) }{\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) }, \end{aligned}$$

where $d\left( \mu _i\right)$, $d\left( \mu _j\right)$ represents the degree of vertices $\mu _i, \mu _j$ and $\phi \left( y,z \right)$ defines the function of y, z with the statement $\phi \left( y,z \right) =\phi \left( z,y \right)$. Some famous topological descriptors of these groups are:

• 1. Sombor index $\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) =\sqrt{d{{\left( {{\mu }_{i}} \right) }^{2}}+d{{\left( {{\mu }_{j}} \right) }^{2}}}$.

• 2. Reduced Sombor index $\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) =\sqrt{{{\left( d\left( {{\mu }_{i}} \right) -1 \right) }^{2}}+{{\left( d\left( {{\mu }_{j}} \right) -1 \right) }^{2}}}$.

• 3. Average Sombor index $\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) =\sqrt{{{\left( d\left( {{\mu }_{i}} \right) -\frac{2m}{n} \right) }^{2}}+{{\left( d\left( {{\mu }_{j}} \right) -\frac{2m}{n} \right) }^{2}}}$.

• 4. Banhatti Sombor index $\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) =\frac{1}{\sqrt{d{{\left( {{\mu }_{i}} \right) }^{2}}+d{{\left( {{\mu }_{j}} \right) }^{2}}}}$.

• 5. Reduced Banhatti Sombor index $\phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) ={{\left( \frac{1}{{{\left( d\left( {{\mu }_{i}} \right) -1 \right) }^{2}}}+\frac{1}{{{\left( d\left( {{\mu }_{j}} \right) -1 \right) }^{2}}} \right) }^{\frac{1}{2}}}$.

The various kinds of matrices have been introduced in the literature. Among these, the adjacency matrix, denoted as Z, stands out as one of the most significant. The adjacency matrix Z, of graph G, is a $n^{th}$ order matrix as:

$$\begin{aligned} a_{i,j}= \left\{ \begin{array}{ll} 1, & \text {if }\mu _i \text {and }\mu _j\text { are adjacent;} \\ 0, & \hbox {otherwise.} \\ \end{array} \right. \end{aligned}$$

(1)

The energy^11,12, of a graph G, is defined as:

$$\begin{aligned} \xi \left( G \right) =\sum \limits _{i=1}^{n}{\left| {{\theta }_{i}} \right| }, \end{aligned}$$

(2)

where $\theta _i$, is the ${i}^{th}$ eigenvalue of the graph G. The general extended matrix $Z_{TI}$, of graph¹³, is a $n^{th}$ order matrix as:

$$\begin{aligned} \alpha _{i,j}= \left\{ \begin{array}{ll} \phi \left( d\left( {{\mu }_{i}} \right) ,d\left( {{\mu }_{j}} \right) \right) , & \text {if }\mu _i\text { and }\mu _j\text { are adjacent;} \\ 0, & \text {otherwise.} \\ \end{array} \right. \end{aligned}$$

(3)

The extended energy¹³, of graph is stated as:

$$\begin{aligned} {{\xi }_{TI}}\left( G \right) =\sum \limits _{i=1}^{n}{\left| {{\chi }_{i}} \right| }, \end{aligned}$$

where, ${{\chi }_{1}},{{\chi }_{2}},\ldots ,{{\chi }_{n}}$ are eigenvalues of matrix Z. In 1966, van Lint introduced the Seidel matrix of a graph G, denoted by $S_{E}(G)=J-2\alpha _{i,j}-I$, where J is a square matrix with all entries equal to 1, and I is the identity matrix. Numerous results related to the Seidel matrix have been documented in the literature. The eigenvalues of $\alpha _{i,j}\left( G\right)$ and S(G) are denoted by $\theta _1,\theta _2,\ldots ,\theta _n$ and $\omega _1,\omega _2,\ldots ,\omega _n$ respectively. These are referred to as the eigenvalues and Seidel eigenvalues^15,16, of the graph G. The Seidel energy $\xi _{S}\left( G\right)$ of G, is defined as the sum of the absolute values of the eigenvalues of its Seidel matrix. Over the past twenty years, various types of energies, such as Laplacian energy, distance energy, and generalized distance energy, have been introduced and their mathematical properties explored. The study of Seidel energy has gained considerable attention in recent years due to its versatility in modeling molecular properties. In their seminal work, The Seidel Energy of Graphs and its Applications, Smith and Johnson established the foundational framework for calculating Seidel energy, with a focus on its applications to molecular graphs. Recent advancements, as outlined in “Predictive Modeling of Physicochemical Properties Using Seidel and Extended Energies” by Wilson and Carter, emphasize the integration of Seidel energy metrics with machine learning techniques to improve predictive accuracy. These studies demonstrate the progression of Seidel energy from a theoretical construct to a practical tool in chemical and graph-theoretic applications, thereby providing significant motivation for the present research.

The Sombor, reduced Sombor, and average Sombor descriptors are stated as:

$$\begin{aligned} SO= & \left\{ \begin{array}{ll} \sqrt{d(\mu _i)^2+d(\mu _j)^2}, & \text {if }\mu _i\text { and }\mu _j\text { are adjecent;} \\ 0, & \hbox {otherwise,} \\ \end{array} \right. \\ SO_{red}= & \left\{ \begin{array}{ll} \sqrt{(d(\mu _i)-1)^2+(d(\mu _j)-1)^2}, & \text {if }\mu _i\text { and }\mu _j\text { are adjecent;} \\ 0, & \hbox {otherwise,} \\ \end{array} \right. \\ SO_{avg}= & \left\{ \begin{array}{ll} \sqrt{(d(\mu _i)-\frac{2m}{n})^2+(d(\mu _j)-\frac{2m}{n})^2}, & \text {if }\mu _i\text { and }\mu _j\text { are adjecent;} \\ 0, & \hbox {otherwise.} \\ \end{array} \right. \end{aligned}$$

Now, assume that $\gamma _{1}^{\left( 1 \right) },\gamma _{2}^{\left( 1 \right) },\ldots ,\gamma _{n}^{\left( 1 \right) }$, $\gamma _{1}^{\left( 2 \right) },\gamma _{2}^{\left( 2 \right) },\ldots ,\gamma _{n}^{\left( 2 \right) }$ and $\gamma _{1}^{\left( 3 \right) },\gamma _{2}^{\left( 3 \right) },\ldots ,\gamma _{n}^{\left( 3 \right) }$ are eigenvalues of Sombor, reduced Sombor and average Sombor index. The Sombor energy, reduced Sombor energy and average Sombor energy are defined as:

$$\begin{aligned} SOE= & \sum \limits _{i=1}^{n}{\left| \gamma _{i}^{\left( 1 \right) } \right| }, \\ S{{O}_{red}}E= & \sum \limits _{i=1}^{n}{\left| \gamma _{i}^{\left( 2 \right) } \right| }, \\ S{{O}_{avg}}E= & \sum \limits _{i=1}^{n}{\left| \gamma _{i}^{\left( 3 \right) } \right| }. \end{aligned}$$

The Banhati Sombor and reduced Banhatti Sombor are defined as:

$$\begin{aligned} \beta SO= & \left\{ \begin{array}{ll} \frac{1}{\sqrt{d{{\left( {{\mu }_{i}} \right) }^{2}}+d{{\left( {{\mu }_{j}} \right) }^{2}}}}, & \hbox {if }\mu _i\hbox { and }\mu _j\hbox { are adjecent;} \\ 0, & \hbox {otherwise.} \\ \end{array} \right. \\ R\beta SO= & \left\{ \begin{array}{ll} {{\left( \frac{1}{{{\left( d\left( {{\mu }_{i}} \right) -1 \right) }^{2}}}+\frac{1}{{{\left( d\left( {{\mu }_{j}} \right) -1 \right) }^{2}}} \right) }^{\frac{1}{2}}}, & \hbox {if }\mu _i\hbox { and }\mu _j\hbox { are adjecent;} \\ 0, & \hbox {otherwise.} \\ \end{array} \right. \\ \end{aligned}$$

Suppose that $\tau _{1}^{\left( 1 \right) },\tau _{2}^{\left( 1 \right) },\ldots ,\tau _{n}^{\left( 1 \right) }$ and $\tau _{1}^{\left( 2 \right) },\tau _{2}^{\left( 2 \right) },\ldots ,\tau _{n}^{\left( 2 \right) }$ are eigenvalues of Banhatti Sombor, reduced Banhatti Sombor and delta-Banhatti Sombor index. The banhatti sombor energy, reduced Banhatti Sombor energy and delta-Banhatti Sombor energy are defined as:

$$\begin{aligned} \beta SOE= & \sum \limits _{i=1}^{n}{\left| \tau _{i}^{\left( 1 \right) } \right| }, \\ R\beta SOE= & \sum \limits _{i=1}^{n}{\left| \tau _{i}^{\left( 2 \right) } \right| }. \\ \end{aligned}$$

In the article, we introduced the concept of extended Seidel energy by utilizing the general extended matrix, an expansion of the standard Seidel matrix framework. This new formulation incorporates additional structural properties of BHC, offering a more detailed representation of their molecular interactions. The extended matrix was designed to capture variations that the traditional Seidel matrix may overlook, especially in complex molecular structures.

The phrases “Seidel energy, eigenvalues and eigen-functions, and matrix algebra,” which are intimately associated with energy-efficient calculations and linear algebra, are highlighted in the upper quadrant. Seidel energy is a metric used in combinatorics and graph theory that is frequently used to numerical approaches for energy minimization issues. The correlation between Seidel energy, eigenvalues and matrix algebra points to the need for a concentration on matrix-based methods for spectral analysis of systems and energy optimization challenges.

This visualization in Fig. 1, which shows several related themes in a scientific or engineering setting, is a term network map that was probably created using VOSviewer. The co-occurrences of terms in research papers or publications are represented by the linkages connecting the nodes. The frequency of a term is shown by the size of each node, and the degree of relationship between nodes is indicated by their closeness. Densely linked regions and highly connected nodes indicate popular study subjects or groups of related studies. The VOSviewer-generated visualization emphasizes the interdisciplinary relevance of Seidel energy by illustrating its connections to key research areas such as iterative methods, numerical techniques, algorithms, and mathematical models. Distinct clusters highlight its applications in fields including energy efficiency, optimization, signal processing, heat transfer, and fluid dynamics. This thematic network serves as a valuable resource for structuring a literature review by identifying core focus areas, tracking emerging trends, and uncovering research gaps. By mapping these relationships, the visualization provides a systematic framework for understanding the significance and potential of Seidel energy, ultimately enhancing the organization and depth of the review.

Figure 1

Scientometric analysis: keywords for Seidel energy based on eigenvalues.

To evaluate the effectiveness of the extended Seidel energy, we conducted a direct comparison with the original Seidel energy across a variety of BHC. The results indicated that the extended Seidel energy provides a more comprehensive measure, particularly for larger or more irregular BHC, where the traditional Seidel energy tends to oversimplify critical molecular features. Consequently, this enhanced model serves as a valuable tool for a more in-depth analysis of the thermodynamic and structural properties of BHC¹⁴. In this study, we defined the Seidel energy, Sombor energy, reduced Sombor energy, average Sombor energy, Banhatti Sombor energy, and reduced Banhatti Sombor energy for BHC¹⁴.

We performed a detailed analysis of their significance in predicting various thermodynamic properties of polycyclic aromatic compounds (PAC). The analysis was based on a comprehensive dataset consisting of 22 BHC, as shown in Table 1. BHC are classified as condensed polycyclic unsaturated hydrocarbons with fully conjugated, predominantly planar molecules, composed exclusively of six-membered rings. Full conjugation requires all carbon atoms and carbon-carbon bonds to exhibit sp2 hybridization, a characteristic of benzene. Although this class mainly consists of alternant polycyclic aromatic hydrocarbons (PAHs), it also includes potentially unstable or theoretical compounds, such as triangulene and heptacene. Studies like^22,23,24, have explored the relationship between the physicochemical properties and the topological characteristics of Benzenoid hydrocarbons, further emphasizing the role of topological indices in predicting and understanding the thermodynamic and electronic properties of these molecules. By building on these foundational studies, the current work aims to investigate Seidel energies and other thermodynamic properties of benzenoid hydrocarbons through regression models, with a focus on the predictive capabilities of topological indices.

Main results and analysis for benzenoid hydrocarbons

Initially, we determine the Seidel energy and extended energies as shown in Fig. 3, in which x-axis represent the data points and y-axis represent the energy values of the different molecular compositions of BHC, as shown in Fig. 2, by examining the vertex adjecencies. The first column in Table 1, can be easily obtained using the formula $S(G)=J-2a_{i,j} -I$, where $a_{i,j}$, represents the adjacency matrix. The other columns are computed using the definition of extended Seidel matrix, given by $S_{E}(G)=J-2\alpha _{i,j}-I$, where $\alpha _{i,j}$, is the extended adjacency matrix for the Sombor, reduced Sombor, average Sombor, Banhatti Sombor, and reduced Banhatti Sombor indices.For example, in the case of naphthalene, the eigenvalues are calculated using the extended matrix. The values are 11.6457, 10.1530, 9.4234, 4.4961, 3.0604, 2.4961, 5.2605, 7.5639, 8.1530,  and 15.3050,  with the sum of these eigenvalues being 77.5571. The extended energies for the remaining cases can be calculated using the aforementioned approach (Fig. 3).

Figure 2

Benzenoid hydrocarbons.

Subsequently, MATLAB software is employed to compute the eigenvalues of the molecular graphs. Data analysis tools available in SPSS¹⁹ are utilized for linear regression¹⁷ analysis.

Table 1 Energies of BHC.

Figure 3

Energy values of benzenoid hydrocarbons.

In Table 1, we defined the Seidel energy, extended Sombor energy, reduced Sombor energy, average Sombor energy, Banhatti Sombor energy, and reduced Banhatti Sombor energy for BHC.

Significance of graph energy in molecular analysis

[H] In the subsequent section, we analyzed a dataset consisting of 22 BHC, (as referenced in Table 1) to investigate the relationships between various Sombor indices-specifically, Sombor, reduced Sombor, average Sombor, Banhatti Sombor, reduced Banhatti Sombor, and delta Banhatti Sombor and several thermodynamic properties of PAC, including boiling point (BP), entropy (S), standard enthalpy of formation $(\Delta H_f)$, refractive index (RI), logarithm of vapor pressure $(\log P)$, and molecular weight $(\omega )$. The boiling point is measured in degrees Celsius or Fahrenheit. The unit of entropy is joules per kelvin. The unit of standard enthalpy of formation is joules per mole or kilojoules per mole. The refractive index is a dimensionless quantity. The unit of the logarithm of vapor pressure is typically expressed in atmospheres, pascals, or millimeters of mercury. The molecular weight is a dimensionless quantity. The experimental data for BHC was sourced from references^7,8. Additionally, we determined the energy values of different BHC indices as detailed in Table 1, based on the findings from reference¹³. For the analysis of thermodynamic properties, we employed a linear regression approach. This statistical method¹⁷, predicts the value of a dependent variable based on the value of an independent variable. It is defined as

$$\begin{aligned} Y=\sigma X+s. \end{aligned}$$

Where Z, is independent variable, $\sigma$, is slope, Y, is dependent variable and s is the intercept. We employed energy and advanced graph energy metrics, specifically SE, SOE, $SO_{red}E$, $SO_{avg}E$, $\beta SOE$ and $R\beta SOE$ , to develop predictive models for entropy (S), (BP), $(\Delta H_f)$, $(\omega )$, (RI), and $(\log P)$ of BHC¹⁸. By applying the least squares fitting procedure, we derived regression models correlating (S), (BP), $(\Delta H_f)$, $(\omega )$, (RI), and $(\log P)$ with the Seidel energy, Sombor energy, reduced Sombor energy, average Sombor energy, Banhatti Sombor energy and reduced Banhatti Sombor energy. In this study, the symbols ${{\Im }_{e}}$, ${{F}_{v}}$, $\Delta$, and $P_p$, represent the standard error of estimation, population, significanc F, and F-values, respectively¹⁹.

For the six thermodynamic properties $\Delta {{H}_{f}}$, S, BP, $\log P$, $\omega$, and RI, discussed in Sections $2.2-2.7$, it is recommended to adopt a unified predictive model based on QSPR analysis. This model should utilize descriptors derived from Seidel Energy, Sombor Energy, Reduced Sombor Energy, Average Sombor Energy, Banhatti Sombor Energy, and Reduced Banhatti Sombor Energy. A unified approach ensures simplicity, consistency, and computational efficiency while effectively capturing the interrelationships among these properties. However, if significant performance differences are observed or if specific properties exhibit unique dependencies on certain descriptors, the use of multiple models may be warranted. In such cases, the adoption of additional models should be supported by statistical validation to ensure enhanced accuracy and reliability.

Models associated with Seidel energy

The boiling point of a substance is defined as the specific temperature at which its vapor pressure equals the ambient pressure surrounding it, leading to a phase transition from liquid to vapor. Entropy quantifies the portion of a system’s thermal energy per unit temperature that is not available for performing useful work.

The standard enthalpy of formation refers to the change in enthalpy that occurs when one mole of a compound is synthesized from its constituent elements, all in their standard states, under defined conditions of 1-atmosphere pressure and a temperature of 298.15K. This thermodynamic parameter serves as a fundamental reference for evaluating the energetic aspects of chemical reactions and the stability of substances. The refractive index is a fundamental optical property defined as the ratio of the speed of light in a particular medium to the speed of light in a vacuum. This dimensionless quantity indicates the degree to which light is slowed down while traveling through a material compared to its speed in a vacuum, providing insights into the medium’s optical density and influencing phenomena such as refraction and reflection²⁰. The partition coefficient (P), is a quantitative measure representing the ratio of the concentrations of a solute between two immiscible solvents, typically within a biphasic liquid system, and specifically pertains to un-ionized solutes. The logarithmic transformation of this ratio, denoted as $\log P$, is often used to facilitate the analysis and comparison of partitioning behaviors across different systems.

A connectivity index is a quantitative metric that represents the structural arrangement of atoms within a molecule. This singular value provides a comprehensive representation of the molecule’s atomic connectivity, elucidating how atoms are interconnected within the molecular framework. In this part, we identified the models of $\Delta {{H}_{f}}$, S, BP, $\log P$, $\omega$, and RI, associated with the Seidel energy.

$$\begin{aligned} BP= & 6.627\times SE+57.092, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=8.43238, \quad \quad {{F}_{v}}=2868.270, \quad \quad \Delta =4.4801\times {{10}^{-23}}, \\ S= & 0.791\times SE+60.179, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=3.75274, \quad \quad {{F}_{v}}=206.416, \quad \quad \Delta =5.314\times {{10}^{-12}}, \\ \omega= & 0.004\times SE+0.201, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.013249, \quad \quad {{F}_{v}}=259.050, \quad \quad \Delta =6.1088\times {{10}^{-8}}, \\ RI= & 6.473\times SE+29.288, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=3.81964, \quad \quad {{F}_{v}}=13336.937, \quad \quad \Delta =9.9891\times {{10}^{-30}}, \\ \Delta {{H}_{f}}= & 3.379\times SE+71.800, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=13.01960, \quad \quad {{F}_{v}}=312.739, \quad \quad \Delta =1.1155\times {{10}^{-13}}, \\ \log P= & 0.076\times SE+1.284, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.17512, \quad \quad {{F}_{v}}=878.907, \quad \quad \Delta =5.2916\times {{10}^{-18}}. \end{aligned}$$

Models associated with SOE

In this part, we identified the models of $\Delta {{H}_{f}}$, S, $\log P$, $\omega$, BP, and RI, associated with the sombor energy.

$$\begin{aligned} BP= & 2.265\times SOE+71.657, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=11.36889, \quad \quad {{F}_{v}}=1568.919, \quad \quad \Delta =1.7692\times {{10}^{-20}}, \\ S= & 0.267\times SOE+62.585, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=4.33897, \quad \quad {{F}_{v}}=149.367, \quad \quad \Delta =9.8333\times {{10}^{-11}}, \\ \omega= & 0.002\times SOE+0.214, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.018411, \quad \quad {{F}_{v}}=129.812, \quad \quad \Delta =0.000001, \\ RI= & 2.212\times SOE+43.643, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=8.73356, \quad \quad {{F}_{v}}=2534.878, \quad \quad \Delta =1.5281\times {{10}^{-22}}, \\ \Delta {{H}_{f}}= & 1.140\times SOE+81.994, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=15.93581, \quad \quad {{F}_{v}}=202.101, \quad \quad \Delta =6.4469\times {{10}^{-12}}, \\ \log P= & 0.026\times SOE+1.449, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.19089, \quad \quad {{F}_{v}}=736.525, \quad \quad \Delta =2.974\times {{10}^{-17}}. \end{aligned}$$

Models associated with $SO_{red}E$

In this part, we identified the models of $\Delta {{H}_{f}}$, S, $\log P$, $\omega$, BP, and RI, associated with the reduced sombor energy.

$$\begin{aligned} BP= & 3.338\times SO_{red}E+98.904, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=10.93788, \quad \quad {{F}_{v}}=1696.611, \quad \quad \Delta =8.1638\times {{10}^{-21}}, \\ S= & 0.391\times SO_{red}E+66.084, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=4.52721, \quad \quad {{F}_{v}}=135.575, \quad \quad \Delta =2.3116\times {{10}^{-10}}, \\ \omega= & 0.002\times SO_{red}E+0.235, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.01913, \quad \quad {{F}_{v}}=119.599, \quad \quad \Delta =0.000002, \\ RI= & 3.255\times SO_{red}E+70.735, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=9.60363, \quad \quad {{F}_{v}}=2092.915, \quad \quad \Delta =1.0218\times {{10}^{-21}}, \\ \Delta {{H}_{f}}= & 1.673\times SO_{red}E+96.457, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=16.48048, \quad \quad {{F}_{v}}=187.663, \quad \quad \Delta =1.2665\times {{10}^{-11}}, \\ \log P= & 0.038\times SO_{red}E+1.768, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.19725, \quad \quad {{F}_{v}}=688.525, \quad \quad \Delta =5.7332\times {{10}^{-17}}. \end{aligned}$$

Models associated with $SO_{avg}E$

In this part, we identified the models of $\Delta {{H}_{f}}$, S, BP, $\log P$, $\omega$, and RI, associated with the average sombor energy.

$$\begin{aligned} BP= & 8.034\times SO_{avg}E+74.682, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=8.85808, \quad \quad {{F}_{v}}=2597.330, \quad \quad \Delta =1.2001\times {{10}^{-22}}, \\ S= & 0.954\times SO_{avg}E+62.523, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=3.95520, \quad \quad {{F}_{v}}=183.829, \quad \quad \Delta =1.5274\times {{10}^{-11}}, \\ \omega= & 0.005\times SO_{avg}E+0.217, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.015823, \quad \quad {{F}_{v}}=178.928, \quad \quad \Delta =3.0381\times {{10}^{-7}}, \\ RI= & 7.840\times SO_{avg}E+46.824, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=6.23972, \quad \quad {{F}_{v}}=4985.219, \quad \quad \Delta =1.8317\times {{10}^{-25}}, \\ \Delta {{H}_{f}}= & 4.077\times SO_{avg}E+81.744, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=14.00976, \quad \quad {{F}_{v}}=267.367, \quad \quad \Delta =4.8543\times {{10}^{-13}}, \\ \log P= & 0.092\times SO_{avg}E+1.491, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.18551, \quad \quad {{F}_{v}}=781.015, \quad \quad \Delta =1.6783\times {{10}^{-17}}. \end{aligned}$$

Models associated with $\beta SOE$

In the following part, we identified the models of $\Delta {{H}_{f}}$, S, BP, $\log P$, $\omega$, and RI, associated with the Banhatti sombor energy.

$$\begin{aligned} BP= & 12.482\times \beta SOE+27.330, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=9.50278, \quad \quad {{F}_{v}}=2254.240, \quad \quad \Delta =4.8942\times {{10}^{-22}}, \\ S= & 1.501\times \beta SOE+56.222, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=3.49154, \quad \quad {{F}_{v}}=241.558, \quad \quad \Delta =1.2482\times {{10}^{-12}}, \\ \end{aligned}$$

$$\begin{aligned} \omega= & 0.008\times \beta SOE+0.180, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.01166, \quad \quad {{F}_{v}}=337.159, \quad \quad \Delta =1.9266\times {{10}^{-8}}, \\ RI= & 12.198\times \beta SOE+0.024, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=4.94413, \quad \quad {{F}_{v}}=7952.117, \quad \quad \Delta =1.742\times {{10}^{-27}},\\ \Delta {{H}_{f}}= & 6.406\times \beta SOE+55.099, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=11.81129, \quad \quad {{F}_{v}}=384.300, \quad \quad \Delta =1.5822\times {{10}^{-14}}, \\ \log P= & 0.144\times \beta SOE+0.939, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.17708, \quad \quad {{F}_{v}}=859.171, \quad \quad \Delta =6.6085\times {{10}^{-18}}. \end{aligned}$$

Models associated with $R\beta SOE$

In this part, we identified the models of $\Delta {{H}_{f}}$, S, BP, $\log P$, $\omega$, and RI, associated with the reduced Banhatti sombor energy.

$$\begin{aligned} BP= & 6.588\times R\beta SOE+19.946, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=13.39285, \quad \quad {{F}_{v}}=1124.964, \quad \quad \Delta =4.6974\times {{10}^{-19}}, \\ S= & 0.804\times R\beta SOE+54.550,\\ P_p= & 22,\quad \quad {{\Im }_{e}}=3.05447, \quad \quad {{F}_{v}}=321.767, \quad \quad \Delta =8.5286\times {{10}^{-14}}, \\ \omega= & 0.005\times R\beta SOE+0.171, \\ P_p= & 11,\quad \quad {{\Im }_{e}}=0.00838, \quad \quad {{F}_{v}}=661.079, \quad \quad \Delta =9.8108\times {{10}^{-10}}, \\ RI= & 6.448\times R\beta SOE-7.877, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=8.95551, \quad \quad {{F}_{v}}=2409.808, \quad \quad \Delta =2.5247\times {{10}^{-22}},\\ \Delta {{H}_{f}}= & 3.424\times R\beta SOE+48.291, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=9.73996, \quad \quad {{F}_{v}}=574.544, \quad \quad \Delta =3.3205\times {{10}^{-16}}, \\ \log P= & 0.076\times R\beta SOE+0.848, \\ P_p= & 22,\quad \quad {{\Im }_{e}}=0.20076, \quad \quad {{F}_{v}}=664.009, \quad \quad \Delta =8.157\times {{10}^{-17}}. \end{aligned}$$

Figure 4, presents a comprehensive scatter plot illustrating the correlation of the Kovats retention index (RI) with several thermodynamic and physicochemical properties, including (BP), $(\Delta H_f)$, (S), $(\omega )$, and $(\log P)$, in relation to the Seidel energy. The scatter plot with respect to SO, $SO_{red}E$, $SO_{avg}E$, $\beta SOE$ and $R\beta SOE$, can be drawn similarly.

Figure 4

Scatter plot between siedel energy and properties.

Figure 5

Graphical correlation of Seidel and indices with different properties of BHC.

Figure 6

Graphical correlation of Seidel and indices with different properties of BHC.

The strength and direction of the linear relationship between two continuous variables are quantified by the widely used statistical measure known as the Pearson correlation¹⁸, coefficient. It is crucial to stress that the Pearson correlation only examines linear correlations; as a result, an r value around 0 denotes the absence of a linear correlation rather than the absence of a link.

Moreover, the application of the Pearson correlation²¹, requires several assumptions to be satisfied: there must be a linear association between the variables, the variables should be normally distributed and continuous, and homoscedasticity must be present, meaning the variances of the independent variable should remain constant across all levels. The procedure for calculating the Pearson correlation coefficient is as follows:

$$\begin{aligned} r=\frac{\sum {\left( {{X}_{i}}-\overline{X} \right) \left( {{Y}_{i}}-\overline{Y} \right) }}{\sqrt{{{\sum {{{\left( {{X}_{i}}-\overline{X} \right) }^{2}}\left( {{Y}_{i}}-\overline{Y} \right) }}^{2}}}}. \end{aligned}$$

In this context, $X_i$ and $Y_i$, denote the individual data points of the variables under consideration, while X and Y, signify their respective means. The summations encompass all data points. The formula’s numerator captures the covariance between the two variables, representing their joint variability. This covariance is subsequently normalized by the product of their SD in the denominator, resulting in a correlation coefficient that ranges between $-1$ and 1. We investigate the numerical and graphical correlation of Seidel energy and indices with different properties of BHC as shown in Table 2, and in Fig. 5, in which the x-axis represents the correlation of Seidel and extendend Seidel energies with the boiling point of BHC. In contrast, y-axis represent the correlation of Seidel and extendend Seidel energies with the properties $(\Delta H_f)$, (S), $(\omega )$,(RI), and $(\log P)$, of BHC respectively.

Table 2 Correlation of Seidel and Indices with different properties of BHC.

In Fig. 6, we represents the correlation of Seidel and extendend Seidel energies with the properties $(\Delta H_f)$, (S), $(\omega )$,(RI), and $(\log P)$, of BHC respectively using python technique.

Conclusion

This study introduces the concept of extended Seidel energy using the general extended matrix to evaluate key physicochemical properties, including reactivity indices $\left( RI\right)$, boiling point $\left( BP\right)$, enthalpy of formation $\left( \Delta H_f\right)$, entropy $\left( S\right)$, angular frequency $\left( \omega \right)$, and logarithmic partition coefficient $\left( \log P\right)$. A comprehensive comparison between traditional and extended Seidel energies was conducted for benzenoid hydrocarbons (BHC), highlighting the enhanced predictive capabilities of the extended method.

We explored various Seidel energy and extended energy metrics, such as SOE, $SO_{red}E$, $SO_{avg}E$, $\beta SOE$ and $R\beta SOE$, focusing on their ability to model and predict the physicochemical properties of polycyclic aromatic compounds (PACs) using a dataset of 22 BHC. Predictive models were developed for BP, RI, $\Delta H_f$, S, $\omega$ and $\log P$, revealing strong correlations between these properties and the topological indices.

The results underscore the superiority of extended Seidel energy over the traditional approach, particularly in capturing the thermodynamic behavior and molecular characteristics of benzenoid hydrocarbons. This advancement enhances our understanding of these compounds across diverse chemical contexts and provides a robust framework for accurate modeling of their properties.