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This article aims to explore two emerging areas of graph theory: chemical graph theory and domination theory. We specifically focus on a graph parameter that combines aspects of graph energy and domination, known as the dominating energy of a simple connected graph. This concept refers to the sum of the absolute values of the eigenvalues of the corresponding dominating matrix. Additionally, we discuss several variants of dominating energy, including total dominating energy, connected dominating energy, and a distance-based variant called hop dominating energy. We also introduce the Estrada version and Resolvent version of these dominating energies and examine their predictive capabilities in relation to a set of 12 physico-chemical properties and the -electron energy of 35 benzenoid hydrocarbons through linear, quadratic, cubic, and multiple linear regression analyses. As a result, we identify the best predictive models, which have applications in various fields, including health science, pharmaceuticals, production, and design engineering. These best predictive models are tested using k-fold cross-validation to assess their stability. Furthermore, a comparative study is conducted to evaluate how the regression models based on domination-based energy parameters perform against models derived from well-known degree-based topological indices, such as the first and second Zagreb indices, the Forgotten index, the Randic index, and the Wiener index. This analysis aims to demonstrate that the domination-based energy parameters are more effective in predicting the physicochemical properties of the selected benzenoid hydrocarbons. Additionally, an external validation is also performed using another set of 5 benzenoid hydrocarbons to ensure the efficacy of the best-fitting models based on domination parameters.
Keywords:-electron energy, Dominating energy, Total dominating energy, Connected dominating energy, Hop dominating energy, Estrada index, Resolvent energy
Subject terms: Chemistry, Mathematics and computing
Introduction
Domination in graphs is one of the prominent areas of graph theory that has been well studied. A subset of vertices of a graph is said to form a dominating(total dominating) set of G if each vertex in (in G) is adjacent to a vertex in S. The least cardinality of a minimal dominating set (total dominating set) is called the domination number (total domination number) of G and is indicated by (), respectively. A dominating set S of a graph G is a connected dominating set if , the subgraph induced by S, is connected. The minimum cardinality of the connected dominating set is the connected domination number and is denoted by . A subset of a graph G is called a hop dominating set if for every , there exists a vertex u in S such that in G. The least cardinality of a minimum hop dominating set is called the hop domination number of G and denoted by . In literature, there are several works done effectively in this area, and newer versions of dominations have been introduced as and when a necessity arises in areas related to Engineering, Medical Science, Optimization Problems, Social Media network analysis, etc., For a detailed study on dominations and its variants, one may refer1,2.
The energy of a graph G is defined as the sum of absolute values of the eigenvalues of the (0, 1)-adjacency matrix corresponding to a graph G with n vertices, namely as
. If the eigenvalues of a molecular graph (of a conjugated -electron system) are labeled as , then the total -electron energy (of the underlying molecule in its ground electronic State), as calculated within the Hückel molecular orbital (HMO) approximation (Refer 1, 2), is equal to
1
where and are constants, n is an odd number. If n is an even number, then
Composed solely of six-membered rings, polycyclic aromatic hydrocarbons (PAHs) compounds-also referred to as benzenoid hydrocarbons (BHCs)-are fully conjugated condensed polycyclic unsaturated hydrocarbons that need a high level of chemical and thermodynamic stability. Numerous benzenoid hydrocarbons, including phenanthrene, anthracene, chrysene, pyrene, perylene, benzo[e]pyrene, benzo[a]pyrene, coronene, benzo[ghi]perylene, and anthanthrene, are found in soils and the earliest marine sediments. Certain chemicals, including phenanthrene, perylene, and anthracene, were discovered in a 150-million-year-old fossil sea lily. The chemical industry uses anthracene and naphthalene extensively. Catalytic gas-phase oxidation transforms about 65% of it into phthalic anhydride, which is utilised in the manufacturing of plastics. Zone refining is used to create anthracene, which is employed as a scintillation counter in nuclear physics. Additionally, anthracene has been proposed for use as a semiconductor and photoconductor. Benzo[a]pyrene and numerous other identified carcinogens that are not PAHs are among the more than 3800 chemicals found in tobacco smoke. When tobacco smoke is inhaled, these chemicals instantly invade the lungs. Pyrene is used to make dyes. Chrysene has been used in UV filters and as a photosensitizer. For looking into the applications of Various PAHs (See Ref4).
QSPR (Quantitative Structure-Property Relationship) analysis of drugs and chemical compounds has garnered significant interest from researchers worldwide. Consequently, there are over 500 research articles available on this topic in the literature. Recently, domination-based or domination degree-based QSPR analysis has attracted attention from researchers in the field of chemical graph theory. The following provides a brief survey of this topic.
Hayat et al.5 compared the -electron energy and the well-performing valency-based topological descriptors of lower benzenoid hydrocarbons. Also, Hayat et al.6 examined topological descriptors based on degree to estimate the thermodynamic properties of benzenoid hydrocarbons. Shiv Kumar et al.7 analyzed energy and Estarda index of benzenoid hydrocarbons with several topological indices. Khan et al.8 conducted a comparative study of seven domination parameters alongside the physico-chemical properties of benzenoid hydrocarbons. Kuriachan et al.9 examined the chemical significance of domination and power domination numbers through QSPR analysis. Shashidhara et al.10 investigated the role of domination-based indices in predicting the physico-chemical properties of butane derivatives. Bommahalli Jayaraman et al.11 explored the properties of anti-tuberculosis drugs using QSPR graph modeling and various domination-based topological indices. Lastly, Kuriachan et al.12 predicted the -electron energy and a set of physico-chemical properties of benzenoid hydrocarbons by employing domination degree-based entropies.
In this work, we have considered several variants of dominating energies like the total dominating energy arising from total dominating sets (introduced by Cockayne, Dawes, and Hedetniemi13), connected dominating energy arising from connected dominating sets (introduced by Sampathkumar and Walikar14), and hop dominating energy arising from hop dominating sets (Natarajan and Ayyaswamy15) for a selected set of 35 benzenoid hydrocarbons.
The dominating matrix with respect to a minimum dominating set D of a graph G is defined as a square matrix of order n, where
The energy obtained from this matrix is referred to as dominating energy, which was introduced by Kamal Kumar16. The total dominating energy introduced by Malathy17 is a variant of dominating energy with respect to a minimum total dominating set of G. A distance-based dominating energy parameter known as the hop dominating energy was introduced by Palani18, who explored several bounds related to it.
In the area of topological indices, one of the well-studied indices is the Estrada index of a graph or network G, denoted by EE(G), which is defined as the sum of the exponentials of the eigenvalues of the adjacency matrix of G. Ernesto Estrada introduced this parameter as a measure of the degree of folding of a protein19. Ivan Gutman et al.20 introduced an energy variant named resolvent energy of a graph, defined as and established some bounds on it. At the very outset, we introduce the Estrada version and Resolvent version of the four aforesaid dominating energies and perform a QSPR analysis to identify the best predictors for 12 chosen physico-chemical properties and the -electron energy of 35 benzenoid hydrocarbons.
Molecular structure of some benzenoid hydrocarbons
This section presents the molecular structure graph of a set of 35 benzenoid hydrocarbons (Refer Fig. 1) considered.
A collection of benzenoid hydrocarbons is selected based on a literature review, and their molecular structure are converted into a simple graph G. Then, we obtain various types of matrices associated with molecular graphs, including a dominating matrix , a total dominating matrix , a connected dominating matrix , and a hop dominating matrix , each defined by its minimal set. Next, we calculate the eigenvalues, or latent values, of these matrices, which allows us to compute their respective energies: Dominating Energy (DE), Total Dominating Energy (TDE), Connected Dominating Energy (CDE), and Hop Dominating Energy (HDE). It is important to note that for any simple and connected graph G, there can be multiple minimal dominating sets (as well as variants of these sets), and determining them is an NP-complete problem.
The energy parameters based on domination used in this study are derived from the molecular graph representation of each compound. This representation captures both the topological and energetic characteristics of the molecular structure. These parameters quantify the dominant influence that specific atoms or substructures have on the molecular energy framework, reflecting the balance among connectivity, bond strength, and electronic distribution. Since some physico-chemical properties are determined by these same structural and electronic factors, the correlations observed between domination-based energy parameters and physicochemical properties are meaningful. Therefore, the Quantitative Structure-Property Relationship (QSPR) relationships developed in this study are not merely empirical; they are grounded in the intrinsic connection between molecular topology, electronic interactions, and macroscopic physico-chemical behavior. This provides a mechanistic rationale for the predictive capability of domination-based descriptors within the QSPR framework.
Let us consider a molecular graph G of Naphthalene (Ref. Fig. 2) having domination number of 3 with a minimal dominating set as whose dominating adjacency matrix as
using this matrix, the respective dominating energy (DE) is calculated as 14.14553. Similarly, its total domination number is 6 with a minimal total dominating set
having TDE as 14.29253, with a minimal connected dominating set as having and with a minimal hop dominating set as having . With the help of MATLAB Software, we compute the eigenvalues and their respective energies of a 35 benzenoid hydrocarbons mentioned in Section 2.
Further to the computation of dominating energies, motivated by the idea of Estrada index, we introduce the Estrada-type energy variants, namely
Estrada Dominating Energy(EDE), Estrada Total Dominating Energy(ETDE), Estrada Connected Dominating Energy(ECDE), and Estrada Hop Dominating Energy(EHDE). For the sake of brevity, let us see the precise definition of the Estrada dominating energy of a graph as follows:
The Estrada dominating energy (EDE) of a graph G is the sum of the exponential of the eigenvalues of a dominating matrix of G concerning a minimal dominating set. That is, , where , are the eigen values of the dominating matrix of G. Similarly, the Estarda version of total dominating energy, connected dominating energy, and hop dominating energy can be defined. Emboldened by the idea of resolvent energy given by Ivan Gutman, we introduce a resolvent version of these dominating energies. For instance, we give the definition of resolvent dominating energy as follows: , where , are the eigenvalues of the dominating matrix of G. Similarly, the resolvent total dominating energy (TDER), resolvent connected dominating energy (CDER), and resolvent hop dominating energy (HDER) can be defined.
Finding all eigenvalues (respectively energies) in a graph with more than 5 nodes manually is challenging. Therefore, the following algorithm 1 is used to compute various domination-based energies of 35 BHCs, which has a total running time of , where n is the order of a graph, which includes plotting of the graph and computing various domination-based energies.
After obtaining the computed values of the domination-based energies for 35 benzenoid hydrocarbons, we will proceed to regression analysis to identify the most effective predictors for the selected physico-chemical properties and the -electron energy of these benzenoid hydrocarbons. To achieve this, we will utilize the linear, quadratic, cubic, and multiple linear regression models given in Eqs. (3), (4),(5), (6).
3
4
5
6
We perform linear, quadratic, cubic, and multiple linear regression analyses employing the data analysis tools in Microsoft Excel. To assess the effectiveness of the parameters, we measure the statistical metrics, the squared correlation coefficient with minimum RMSE, and the level of significance p-value (.
Computation of dominating, Estrada dominating and Resolvent dominating energy variants.
Results and discussions
Dominating energies, Estrada versions, Resolvent versions and physico-chemical Properties
This subsection contains the experimental values of some physico-chemical properties of benzenoid hydrocarbons such as total electron energy , Boiling Point (BP), Enthalpy of Vapourization (EV), Flash Point (FP), Molar Refractivity (MR), LogP, Polarizability(P), Molar Volume (MV), Molecular Weight(MW), , Exact Mass(EM), Complexity(C) and Index of Refraction(IR) as shown in Table 4. Dominating Energy(DE), Total Dominating Energy(TDE), Connected Dominating Energy(CDE), and Hop Dominating Energy (HDE) of aforesaid 35 benzenoid hydrocarbons, their Estrada and Resolvent versions are given in Table 1, Table 2, and Table 3.
Table 4.
Experimental values of physico-chemical properties of benzenoid hydrocarbons.
Using the linear regression model given in Eq. (3), we determine the linear relation between 12 physico-chemical properties as well as the -electron energy and 12 dominating energy parameters.
Based on the data provided in Tables 1, 2, 3, and 4, we obtain values for the linear regression model, and the values are listed in Table 5. Note that the maximum obtained against each property is displayed in bold.
Table 5.
The square of the correlation coefficient obtained by the linear regression model.
In the linear regression model, the properties -electron energy , MR, P, MW, and EM are notably well predicted by DE , whereas properties like BP, EV, FP and MV are predicted by Estarda version dominating parameter ETDE. The most suitable predictor for the properties and C is the dominating energy parameter HDE and CDE. The other two properties, LogP and IR, are not predicted by any of these 12 dominating parameters in the linear regression model.
7
8
9
10
11
12
13
14
15
16
17
Quadratic regression analysis
Using the quadratic regression model described in Eq. 4, we conduct a comprehensive analysis of 12 physico-chemical properties alongside the -electron energy of 35 benzenoid hydrocarbons. Table 7 presents the values for each dominating energy parameter, illustrating their predictive accuracy for the properties. The parameters with the highest values are highlighted.
Table 7.
The square of the correlation coefficient obtained by the quadratic regression model.
In the quadratic regression model, the parameter DE is a better predictor for the properties , MR, P, MW and EM. Additionally, the parameter ETDE serves as the best predictor for the properties BP, EV, FP, and MV. Meanwhile, the parameter CDE provides a better prediction for the properties and C. The other two properties, LogP and IR, are not predicted in quadratic analysis by any of these 12 dominating parameters.
18
19
20
21
22
23
24
25
26
27
28
Cubic regression analysis
Using the cubic regression model described in Eq. 5, we present an extensive analysis of the physico-chemical characteristics of 12 and the electron energy of 35 benzenoid hydrocarbons. The values for each dominating energy parameter are shown in Table 9, demonstrating the precision with which they predict properties. The parameters with the highest values are highlighted.
Table 9.
The square of the correlation coefficient obtained by the cubic regression model.
In cubic regression analysis, the parameter DE is a better predictor for the properties , MR, P, MW and EM. Additionally, the parameter ETDE serves as the best predictor for the properties BP, EV, FP and MV. Meanwhile, the parameters HDE and CDE provide better predictions for the properties and C. Notably, the property LogP is predicted only on this cubic regression model.
Based on the results from the linear, quadratic and cubic regression analyses presented in Tables 6, 8 and 10, we observe that the parameter DE effectively predicts the properties , MR, P, MW and EM while CDE serves as the best predictor for the property C in the linear, quadratic and cubic regression model. Additionally, in the linear and cubic regression model, HDE is the best predictor for the property , whereas LogP is predicted in the cubic regression model alone by a dominating energy parameter CDE. Furthermore, the dominating energy parameter ETDE most accurately predicts the properties BP, EV, FP, and MV. These optimal predictive fits resulting from our analyses are illustrated in Figs. 3, 4, and 5.
Table 6.
Best predictive fits from linear regression model.
Cubic regression curves for MR and P against DE, BP and EV against ETDE and LogP against CDE.
Multiple linear regression analysis
In this subsection, we extend our analysis from linear regression to multiple linear regression, examining the effectiveness of the selected parameters in predicting the 13 properties under consideration.
41
42
43
44
45
46
47
48
49
50
51
52
The statistical measures obtained from the multiple linear regression analysis are summarized in Table 11.
In this section, we perform a QSPR analysis using well-known topological indices, namely the first and second Zagreb indices, forgotten index (F index), Randi index and Wiener index. The topological indices considered in our study are presented in the following Table 12.
For brevity, only the results from the quadratic and cubic regression models are presented in the following discussion, despite conducting linear, quadratic, and cubic regression analyses (Refer Table 13, Table 15).
Table 13.
The square of the correlation coefficient obtained by the quadratic regression model for topological indices.
The properties C and MV are better predicted by the parameters and W, while the parameter R predicts all other properties in quadratic regression analysis (Refer Tabel 14).
53
54
55
56
57
58
59
60
61
62
63
Table 14.
Best predictive fits from quadratic regression model for topological indices.
In cubic regression analysis, the parameter is a better predictor for the properties MR, P, and C. Additionally, the parameter W serves as the best predictor for the properties BP, EV, FP and MV. Meanwhile, the parameter R provides better prediction for the properties , MW, , and EM. Notably, the property LogP is predicted only on this cubic regression model by the parameter F (Refer Table 16).
Table 16.
Best predictive fits from cubic regression model for topological indices.
In comparing our results from the cubic regression model and the quadratic regression model, we found that the cubic regression model is the best fit for predicting nearly all the selected physico-chemical properties of BHCs. We will validate these best-fit results using k-fold cross-validation in the following sub section.
k-fold cross-validation
In this section, we perform a k-fold cross-validation using Algorithm 2 to examine the stability of the best predictive fits identified in regression models resulting from domination-based energy parameters as well as the topological indices, whose results are tabulated in Table 17 and Table 18.
Table 17.
5-fold cross-validation based on the best predictive model for Dominating energies.
Polynomial Regression with k-fold cross-validation.
From the 5-fold cross-validation summarised in Table 17 and Table 18, the risk of overfitting is observed for the property LogP, since .
External validation
To test the effectiveness of the best predictive models obtained in this study, we have done an external validation using 5 other benzenoid hydrocarbons, namely Benzo[b]chrysene(), Benzo[e]naphtho[2,3-a]pyrene(), Benzo[a]tetracene(), Dibenzo[a,c]anthracene(), and Dibenzopentaphene(). Their actual and predicted values, with their absolute residual error, are presented in Tables 19, 20 and 21.
Table 19.
Comparison of actual and predicted values of selected benzenoid hydrocarbons.
Dominating energy and its variants are emerging energy parameters that many researchers are currently investigating. Although calculating the domination number of a graph is an NP-complete problem, domination-based energies play a crucial role in predicting the physico-chemical properties of chemical compounds.
In this study, we conducted a regression analysis to identify the best predictors for twelve selected physico-chemical properties, as well as the -electron energy, of a set of 35 benzenoid hydrocarbons (BHCs). We utilized a set of 12 domination-based energy parameters for this analysis. Additionally, we performed another regression analysis using 5 well-known topological indices to compare the effectiveness of the domination-based energy parameters in predicting the physico-chemical properties of BHCs. To ensure the reliability of the regression models resulting from these analyses, we implemented 5-fold cross-validation, achieving the following results:
(i)
The parameter showed good correlation with the property in linear regression.
(ii)
In quadratic regression, the energy parameters DE was found to be the most accurate predictor for the property . The topological index R served as the best predictor for the property .
(iii)
In cubic regression, the dominating energy parameter DE demonstrated better certainty with the properties and , where the energy parameter ETDE outperformed all other selected parameters to predict the properties and . The index W was identified as the most accurate predictor for the properties and . Meanwhile, the index R served as a best predictor for the properties and
(iv)
In Multiple linear regression, we observe that all other properties have been well predicted except the property LogP.
The limitation of having a unique minimum dominating set and its variant is extremely challenging because multiple minimum dominating sets are possible due to its NP-complete nature, resulting in little variations in their respective eigenvalues and their energy values too. However, a sufficient attempt is initiated in this study by making use of a minimum dominating set (and its variants) for a precise regression analysis. The methods employed in this work will help chemists and researchers in making a rational inference on predicting physico-chemical properties of other classes of chemical compounds or drugs to provide efficient treatment for diagnosing diseases, through different dominating energies. The outcomes of this study offer an excellent foundation for future developments and aid in the development of realistic models. This study effectively illustrated how well different regression models predict key physico-chemical properties of various benzenoid hydrocarbons, which is validated for 5 other external benzenoid hydrocarbons, namely Benzo[b]chrysene, Benzo[e]naphtho[2,3-a]pyrene, Benzo[a]tetracene, Dibenzo[a,c]anthracene, and Dibenzopentaphene. Further research could investigate the incorporation of a wider range of molecular descriptors beyond these dominating energies in order to further improve the accuracy of these predictions. This may lead to deeper understandings of QSPR analysis and possibly contribute to more accurate and reliable predictive models.
Author contributions
N.C conceptualized the key idea of this research work. S.S wrote the manuscript and did the analysis part. N.C revised the manuscript and fine-tuned it in its present form, and also N.C supervised the overall progress of the work.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
-electron energy of 35 benzenoid hydrocarbons through linear, quadratic, cubic, and multiple linear regression analyses. As a result, we identify the best predictive models, which have applications in various fields, including health science, pharmaceuticals, production, and design engineering. These best predictive models are tested using k-fold cross-validation to assess their stability. Furthermore, a comparative study is conducted to evaluate how the regression models based on domination-based energy parameters perform against models derived from well-known degree-based topological indices, such as the first and second Zagreb indices, the Forgotten index, the Randic index, and the Wiener index. This analysis aims to demonstrate that the domination-based energy parameters are more effective in predicting the physicochemical properties of the selected benzenoid hydrocarbons. Additionally, an external validation is also performed using another set of 5 benzenoid hydrocarbons to ensure the efficacy of the best-fitting models based on domination parameters.
-electron energy, Dominating energy, Total dominating energy, Connected dominating energy, Hop dominating energy, Estrada index, Resolvent energy
of vertices of a graph 
is said to form a dominating(total dominating) set of G if each vertex in 
(in G) is adjacent to a vertex in S. The least cardinality of a minimal dominating set (total dominating set) is called the domination number (total domination number) of G and is indicated by 
(
), respectively. A dominating set S of a graph G is a connected dominating set if 
, the subgraph induced by S, is connected. The minimum cardinality of the connected dominating set is the connected domination number and is denoted by 
. A subset 
of a graph G is called a hop dominating set if for every 
, there exists a vertex u in S such that 
in G. The least cardinality of a minimum hop dominating set is called the hop domination number of G and denoted by 
. In literature, there are several works done effectively in this area, and newer versions of dominations have been introduced as and when a necessity arises in areas related to Engineering, Medical Science, Optimization Problems, Social Media network analysis, etc., For a detailed study on dominations and its variants, one may refer1,2.
. If the eigenvalues of a molecular graph (of a conjugated 
-electron system) are labeled as 
, then the total 
-electron energy 
(of the underlying molecule in its ground electronic State), as calculated within the Hückel molecular orbital (HMO) approximation (Refer 1, 2), is equal to
and 
are constants, n is an odd number. If n is an even number, then
-electron energy and the well-performing valency-based topological descriptors of lower benzenoid hydrocarbons. Also, Hayat et al.6 examined topological descriptors based on degree to estimate the thermodynamic properties of benzenoid hydrocarbons. Shiv Kumar et al.7 analyzed energy and Estarda index of benzenoid hydrocarbons with several topological indices. Khan et al.8 conducted a comparative study of seven domination parameters alongside the physico-chemical properties of benzenoid hydrocarbons. Kuriachan et al.9 examined the chemical significance of domination and power domination numbers through QSPR analysis. Shashidhara et al.10 investigated the role of domination-based indices in predicting the physico-chemical properties of butane derivatives. Bommahalli Jayaraman et al.11 explored the properties of anti-tuberculosis drugs using QSPR graph modeling and various domination-based topological indices. Lastly, Kuriachan et al.12 predicted the 
-electron energy and a set of physico-chemical properties of benzenoid hydrocarbons by employing domination degree-based entropies.
with respect to a minimum dominating set D of a graph G is defined as a square matrix 
of order n, where
is referred to as dominating energy, which was introduced by Kamal Kumar16. The total dominating energy introduced by Malathy17 is a variant of dominating energy with respect to a minimum total dominating set of G. A distance-based dominating energy parameter known as the hop dominating energy was introduced by Palani18, who explored several bounds related to it.
and established some bounds on it. At the very outset, we introduce the Estrada version and Resolvent version of the four aforesaid dominating energies and perform a QSPR analysis to identify the best predictors for 12 chosen physico-chemical properties and the 
-electron energy of 35 benzenoid hydrocarbons.
, a total dominating matrix 
, a connected dominating matrix 
, and a hop dominating matrix 
, each defined by its minimal set. Next, we calculate the eigenvalues, or latent values, of these matrices, which allows us to compute their respective energies: Dominating Energy (DE), Total Dominating Energy (TDE), Connected Dominating Energy (CDE), and Hop Dominating Energy (HDE). It is important to note that for any simple and connected graph G, there can be multiple minimal dominating sets (as well as variants of these sets), and determining them is an NP-complete problem.
whose dominating adjacency matrix as
having TDE as 14.29253, 
with a minimal connected dominating set as 
having 
and 
with a minimal hop dominating set as 
having 
. With the help of MATLAB Software, we compute the eigenvalues and their respective energies of a 35 benzenoid hydrocarbons mentioned in Section 2.
, where 
, 
are the eigen values of the dominating matrix of G. Similarly, the Estarda version of total dominating energy, connected dominating energy, and hop dominating energy can be defined. Emboldened by the idea of resolvent energy given by Ivan Gutman, we introduce a resolvent version of these dominating energies. For instance, we give the definition of resolvent dominating energy as follows: 
, where 
, 
are the eigenvalues of the dominating matrix of G. Similarly, the resolvent total dominating energy (TDER), resolvent connected dominating energy (CDER), and resolvent hop dominating energy (HDER) can be defined.
, where n is the order of a graph, which includes plotting of the graph and computing various domination-based energies.
-electron energy of these benzenoid hydrocarbons. To achieve this, we will utilize the linear, quadratic, cubic, and multiple linear regression models given in Eqs. (3), (4),(5), (6).
with minimum RMSE, and the level of significance p-value (
.
electron energy 
, Boiling Point (BP), Enthalpy of Vapourization (EV), Flash Point (FP), Molar Refractivity (MR), LogP, Polarizability(P), Molar Volume (MV), Molecular Weight(MW), 
, Exact Mass(EM), Complexity(C) and Index of Refraction(IR) as shown in Table 4. Dominating Energy(DE), Total Dominating Energy(TDE), Connected Dominating Energy(CDE), and Hop Dominating Energy (HDE) of aforesaid 35 benzenoid hydrocarbons, their Estrada and Resolvent versions are given in Table 1, Table 2, and Table 3.
-electron energy and 12 dominating energy parameters.
values for the linear regression model, and the values are listed in Table 5. Note that the maximum 
obtained against each property is displayed in bold.
-electron energy 
, MR, P, MW, and EM are notably well predicted by DE , whereas properties like BP, EV, FP and MV are predicted by Estarda version dominating parameter ETDE. The most suitable predictor for the properties 
and C is the dominating energy parameter HDE and CDE. The other two properties, LogP and IR, are not predicted by any of these 12 dominating parameters in the linear regression model.
-electron energy of 35 benzenoid hydrocarbons. Table 7 presents the 
values for each dominating energy parameter, illustrating their predictive accuracy for the properties. The parameters with the highest 
values are highlighted.
, MR, P, MW and EM. Additionally, the parameter ETDE serves as the best predictor for the properties BP, EV, FP, and MV. Meanwhile, the parameter CDE provides a better prediction for the properties 
and C. The other two properties, LogP and IR, are not predicted in quadratic analysis by any of these 12 dominating parameters.
of 35 benzenoid hydrocarbons. The 
values for each dominating energy parameter are shown in Table 9, demonstrating the precision with which they predict properties. The parameters with the highest 
values are highlighted.
, MR, P, MW and EM. Additionally, the parameter ETDE serves as the best predictor for the properties BP, EV, FP and MV. Meanwhile, the parameters HDE and CDE provide better predictions for the properties 
and C. Notably, the property LogP is predicted only on this cubic regression model.
, MR, P, MW and EM while CDE serves as the best predictor for the property C in the linear, quadratic and cubic regression model. Additionally, in the linear and cubic regression model, HDE is the best predictor for the property 
, whereas LogP is predicted in the cubic regression model alone by a dominating energy parameter CDE. Furthermore, the dominating energy parameter ETDE most accurately predicts the properties BP, EV, FP, and MV. These optimal predictive fits resulting from our analyses are illustrated in Figs. 3, 4, and 5.
against HDE and MV against ETDE.
against DE and FP against ETDE.
value
index and Wiener index. The topological indices considered in our study are presented in the following Table 12.
)
)
and W, while the parameter R predicts all other properties in quadratic regression analysis (Refer Tabel 14).
is a better predictor for the properties MR, P, and C. Additionally, the parameter W serves as the best predictor for the properties BP, EV, FP and MV. Meanwhile, the parameter R provides better prediction for the properties 
, MW, 
, and EM. Notably, the property LogP is predicted only on this cubic regression model by the parameter F (Refer Table 16).
.
), Benzo[e]naphtho[2,3-a]pyrene(
), Benzo[a]tetracene(
), Dibenzo[a,c]anthracene(
), and Dibenzopentaphene(
). Their actual and predicted values, with their absolute residual error, are presented in Tables 19, 20 and 21.
-electron energy, of a set of 35 benzenoid hydrocarbons (BHCs). We utilized a set of 12 domination-based energy parameters for this analysis. Additionally, we performed another regression analysis using 5 well-known topological indices to compare the effectiveness of the domination-based energy parameters in predicting the physico-chemical properties of BHCs. To ensure the reliability of the regression models resulting from these analyses, we implemented 5-fold cross-validation, achieving the following results: 
showed good correlation with the property 
in linear regression.
. The topological index R served as the best predictor for the property 
.
and 
, where the energy parameter ETDE outperformed all other selected parameters to predict the properties 
and 
. The index W was identified as the most accurate predictor for the properties 
and 
. Meanwhile, the index R served as a best predictor for the properties 
and 