Machine learning–based regression analysis of cyclopentane and cyclohexane molecular graphs

Abstract

Cyclopentane is used to make high-performance plastics and composites. Cyclohexane is a polymer that has several specialized applications due to its unique properties, such as good chemical resistance, low moisture absorption, and high thermal stability. Despite their industrial relevance, a systematic comparative analysis of degree-based and neighborhood degree-based topological indices and their predictive capability for chemical properties remains limited in the existing literature. In this study, we compute and analyze several fundamental molecular descriptors, including the Zagreb indices, Randić index, atom bond connectivity index, geometric arithmetic index, sum connectivity index, and augmented Zagreb index for the molecular graphs networks of Cyclopentane and Cyclohexane. These indices can be used to correlate various properties of the polymers with the topological indices. The obtained topological indices are employed to establish quantitative structure–property relationships (QSPR) between molecular structure and physicochemical properties of these polymers. To address the lack of predictive modeling in earlier graph-theoretical studies, a machine-learning-based regression models are developed and applied to evaluate the strength of correlation between molecular descriptors and experimental property data. This approach provides a computationally efficient framework for molecular property estimation and offers a foundation for extending advanced learning models to more complex polymer networks.

Introduction

Triisobutylaluminum-titanium tetra and trichloride catalysts were used to produce Cyclopentane at a conversion rate of 7–14%. Ether was used to extract the unfractionated Cyclopentane, which was found to be crystalline through x-ray examination. However, this method was unable to dissolve 88% of the polymer. After being fully soluble in benzene, the residual material underwent annealing to produce a highly crystalline substance that broke down at temperatures higher than C. The characteristics of this material were acquired by quickly extending a crystalline Cyclopentane fibre from a melt see¹. A triclinic unit cell was used to index crystalline Cyclopentane, with three monomer units within an identity period of 6.6A.

Due to its chemical corrosion resistance and thermal stability, Cyclopentane is utilised in high-performance polymers and composites, which makes it perfect for severe conditions. It is utilised in the medical field to create biocompatible materials for implants and devices due to its inflexibility and compatibility with biological tissues.

Triisobutylaluminum-titanium tetra and trichloride catalysts (4/1) were used to produce Cyclohexane, with conversions restricted only by the reaction conditions. 94% of the unfractionated Cyclohexane was not soluble in ether, and x-ray examination revealed that the substance was crystalline. While some extra benzene was soluble, the majority >90% was only soluble in heated Tetralin. The latter was discovered to be very crystalline and to break down at temperatures higher than C upon annealing. The characteristics of this material were acquired by quickly extending a crystalline Cyclohexane fibre from a melt. Three monomeric units per triclinic unit cell served as the foundation for the indexing of crystalline Cyclohexane, see detail in¹. 6.6A or 13.2A was found to be the identity period along the fibre axis.

While Cyclohexane has good dielectric qualities and is resistant to chemicals and moisture, it is used in protective coatings, encapsulants, and insulating materials. It is appropriate for optical applications like lenses and light guides because of its transparency and stability in a variety of environmental conditions.

In mathematical chemistry, we use mathematical methods to explain and analyse some significant aspects of a chemical structure. This theory works well for creating complicated networks and modelling chemical structures. The study of molecular structure benefits from the application of graph theory to the mathematical modelling of molecular events, which is made possible by chemical graph theory. In^2,3, various graph properties are explained. In graphs, the topological indices shows, the graphical structure and also many other characteristics. Scientific computing, mathematical chemistry, and pharmaceuticals all heavily rely on the topological indices of many graphs are discussed in^4,5.

Let G=(V,E) be a graph with V representing its vertex set and E representing its edge set. The edges of a chemical graph represent the bonds between atoms, and the graph’s vertices represent the atoms. The number of edges that are connected to a vertex r∈ V, represented by , is the degree of r and λ (r) represents neighborhood degree of vertex r∈ V, which is .

Degree-based and neighborhood degree-based topological indices are highly significant and essential to chemical graph theory, especially in the field of chemistry. Amna Amer, M. Irfan and Hamood ul Rehman in⁶ and Amna Amer et. al. in⁷ calculated edge version of degree-based topological indices of boron triangular nanotubes. Neighborhood degree-base topological indices of nanotubes calculated in 2024 are reffered as⁸. We can calculate different formulas by using degree-base topological indices like entropy and M-polynomial. M-Polynomials and entropy based on degree-based topological indices of some graphs calculated in^9,10. These indices can predict various physicochemical properties of chemical compounds, such as heat of formation, strain energy, entropy, enthalpy, and boiling point see the details in^11,12. Hanif et al.¹³ described analysis of two-dimensional nickel-based organometallic network via statistical method, Babar et al.¹⁴ explore the physical properties of polyhydroxybutyrate via eccentricity-based topological indices, Almohanna et al.¹⁵ find topological indices and Gibbs energy for terephthalic tetragonal network via curve-fitting model and Asif et al.¹⁶ explore the work on analysis of cubic zirconia network through eccentricity operators. In¹⁷Irfan et al. Explore Entropy Measures in Polymer Graphs using Logarithmic Regression Model. Comparative entropy analysis of 2D transition metal Tetrahydroxyquinones via machine learning approaches presents in¹⁸

Table 1 and 2 provide a set of topological indices that we will discuss in this article, which includes Zagreb indices, Randić index, atom bond connectivity index, geometric arithmetic index, sum connectivity index, and augmented Zagreb index.

Table 1.

Degree-based topological indices.

S.No.	TIs.	Formula	Authors
1			Gutman and Trinajstić in 197219
2			Gutman and Das in 200420
3			Hao in 201121
4			Bollobás and Erdös in 199822.
5	ABC(G)		Estrada, Torres, Rodríguez, and Gutman in 199823
6	GA(G)		Vukićevič and Furtula in 20093.
7	SCI(G)		Zhou and Trinajstic in 200924
8	AZI(G)		Furtula, Graovac and Vukićeič in 201025

Table 2.

Neighborhood degree-based topological indices.

S.No.	TIs.	Formula	Authors
1			Ghorbani and Hosseinzadeh in 201326
2			Mondal, De and Pal in 201927
3			Verma and Mondal in 201928
4			Verma and Mondal in 201928
5			Ghorbani and Hosseinzadeh in 201029
6			Grovac, Ghorbani, and Hosseinzadeh in 201130.
7			Mondal, Dey, De, and Pal in 202131
8	S(G)		Hosamani in 201732

Computations and graphical analysis of topological indices for cyclopentane

In this part, we present our computation-based results, and we calculate degree-based and neighborhood degree-based topological indices for the Cyclopentane. We have also compared these indices numerically and graphically. The chemical graph of Cyclopentane network with one unit cell and with three unit cells shown in Fig. 1. In the structure of Cyclopentane network, blue vertices represent carbon atoms and red vertices represent hydrogen atoms. In more than one unit of Cyclopentane network, green edges are used for the connection of units. In n units of the Cyclopentane network, the total number of vertices is 19n and the total number of edges is 20n-1. The edge partition of Cyclopentane for degree-based and neighborhood degree-based shown in Table 3 and Table 4. A colorless and flammable liquid, cyclopentane is usually a self-blowing insulation foam blowing agent and a solvent in organic synthesis. Table 5 show physical and chemical characteristics. It is possible to plot these numerical values in a comparative graph in order to show the trends in thermal and physicochemical properties, i.e., correlation between molecular packing (density, molar refractivity) and thermal properties (melting points, boiling points) and hydrophobicity (Log P). These properties are collected from chemical book software.

Fig. 1.

Cyclopentane structure. (a) Chemical Structure of Cyclopentane. (b) Molecular Graph of Cyclopentane for n=1. (c) Molecular Graph of Cyclopentane for n=3.

Table 3.

Degree-based edge partition of cyclopentane for n≥ 2.

	Frequency
(1,3)	3
(1,4)	12n-3
(3,4)	3
(4,4)	8n-4

Table 4.

Neighborhood Degree-base edge partition of cyclopentane for n≥ 3.

	Frequency
(3,6)	2
(3,9)	1
(4,9)	2
(4,10)	10n-4
(4,12)	2
(4,13)	2n-3
(6,12)	1
(9,9)	1
(9,12)	1
(9,13)	1
(10,10)	3n
(10,12)	3
(10,13)	4n-7
(12,13)	1
(13,13)	n-2

Table 5.

Physical and chemical properties of cyclopentane.

Property	Value
Molar weight	96.17
Melting point	–126.5
Boiling point	97
Density	0.704
Flash point	50
Molar refractivity	35.72
Log p	2.363
Polarizability	14.16

• Let G be a graph of the Cyclopentane network from Table 1 and Table 3, we have degree-based topological indices which includes Zagreb indices, Randić index, atom bond connectivity index, geometric arithmetic index, sum connectivity index and augmented Zagreb index are as follows:

• First Zagreb index

  

  
• Second Zagreb index

  

  
• Second modified Zagreb index

  

  
• General Randić index

  

  

  Randić index for particular value at is.

  
• Atom bond connectivity index

  

  
• Geometric arithmetic index

  

  
• Sum connectivity index

  

  
• Augmented Zagreb index

  

  

In Table 6 we have numerical comparison of the indices and their graphs are represented in Fig. 2.

Table 6.

Numerical comparison of degree-based indices of cyclopentane for n=1 to 10.

n				R(G)	ABC(G)	GA(G)	SCI(G)	AZI(G)
1	110	145	4.5	8.09	14.63	16.57	8.07	145.04
2	234	321	9	16.09	29.92	33.97	16.26	325.19
3	358	497	13.5	24.09	45.21	51.37	24.45	505.34
4	482	673	18	32.09	60.5	68.77	32.64	685.49
5	606	849	22.5	40.09	75.79	86.17	40.83	865.64
6	730	1025	27	48.09	91.08	103.57	49.02	1045.79
7	852	1201	31.5	56.09	106.37	120.97	57.21	1225.94
8	978	1377	36	64.09	121.66	138.37	65.4	1406.09
9	1102	1553	40.5	72.09	136.95	155.77	73.59	1586.24
10	1226	1729	45	80.09	152.24	173.17	81.78	1766.39

Fig. 2.

Graphical comparison of degree-based TIs of cyclopentane. (a) Graphical Comparison of M₁(G),M₂(G), GA(G), AZI(G) for Cyclopentane. (b) Graphical Comparison of for Cyclopentane.

• Let G be a graph of the Cyclopentane network, from Table 2 and Table 4, we have neighborhood degree-based topological indices which include neighborhood Zagreb indices, neighborhood Randić index, fourth atom bond connectivity index, the fifth geometric arithmetic index, the second ND index and sanskruti index are as follows:

• Third version of the Zagreb index

  

  
• Neighborhood second Zagreb index

  

  
• Neighborhood second modified Zagreb index

  

  
• Neighborhood general Randić index

  

  

  Randić index for particular value at is

  
• The fourth atom bond connectivity index

  

  
• The fifth geometric arithmetic index

  

  
• The second ND index

  

  
• Sanskruti index

  

  

In Table 7 we have the numerical comparison and the graphical comparison are shown in Fig. 3.

Table 7.

Numerical comparison of neighborhood degree-based indices of cyclopentane for n=1 to 10.

n				NR(G)				S(G)
1	290	1054	0.27	2.82	5.42	17.81	4.37	1403.21
2	642	2547	0.62	5.41	15.23	36.5	9.22	3669.38
3	994	4040	0.97	8	25.04	55.19	14.07	5935.55
4	1346	5533	1.32	10.59	34.85	73.88	18.92	8201.72
5	1698	7026	1.67	13.18	44.66	92.57	23.77	10467.89
6	2050	8519	2.02	15.77	54.47	111.26	28.62	12734.06
7	2402	10012	2.37	18.36	64.28	129.95	33.47	15000.23
8	2754	11505	2.72	20.95	74.09	148.64	38.32	17266.4
9	3106	12998	3.07	23.54	83.9	167.33	43.17	19532.57
10	3458	14491	3.42	26.13	93.71	186.02	48.02	21798.74

Fig. 3.

Graphical comparison of neighborhood degree-based TIs of cyclopentane. (a) Graphical Comparison of for Cyclopentane. (b) Graphical Comparison of for Cyclopentane.

Computations and graphical analysis of topological indices for cyclohexane

In this part, we present our computation-based results, and we calculate degree-based and neighborhood degree-based topological indices for Cyclohexane. We have also compared these indices numerically and graphically. Cyclohexane network of chemical graph and with one unit cell and with three unit cells shown in Fig. 4. In structure of Cyclohexane network blue vertices represent carbon atoms and red vertices represent hydrogen atoms. In more than one unit cell of Cyclohexane network green edge is used for the connection between the units. Cyclohexane network with n units contains a total number of vertices 22n and a total number of edges 23n-1. The edge partition of Cyclohexane for degree-based and neighborhood degree-based are shown in Table 8 and table 9.

Fig. 4.

Cyclohexane Structure. (a) Chemical Structure of Cyclohexane. (b) Molecular Graph of Cyclohexane for n=1. (c) Molecular Graph of Cyclohexane for n=3.

Table 8.

Degree-based Edge partition of cyclohexane for n≥ 2.

	Frequency
(1,3)	3
(1,4)	14n-3
(3,4)	3
(4,4)	9n-4

Table 9.

Neighborhood degree-based edge partition of cyclohexane for n≥ 3.

	Frequency
(3,6)	2
(3,9)	1
(4,9)	2
(4,10)	12n-4
(4,12)	2
(4,13)	2n-3
(6,12)	1
(9,9)	1
(9,12)	1
(9,13)	1
(10,10)	4n
(10,12)	3
(10,13)	4n-7
(12,13)	1
(13,13)	n-2

Cyclohexane, a colorless, flammable liquid with a slight petroleum-like odor, is widely used as a non-polar solvent in chemical processes for the commercial production of adipic acid and caprolactam. Table 10 summarizes the physical and chemical properties. To illustrate patterns in physicochemical and thermal properties, such as the correlation between hydrophobicity (Log P) and thermal properties (melting and boiling temperatures) and molecular packing (density, refractive index), these numerical values can be plotted in a comparative graph.

• Let G be a graph of the Cyclohexane network From Table 1 and Table 8, we have degree-base, topological indices which includes Zagreb indices, Randić index, atom bond connectivity index, geometric arithmetic index, sum connectivity index and augmented Zagreb index are as follows:

Table 10.

Physical and chemical properties of cyclohexane.

Property	Value
Molar weight	110
Melting point	121
Boiling point	128
Density	0.805
Flash point	70
Molar refractivity	36.54
Log p	5
Polarizability	14.49

• First Zagreb index

  

  
• Second Zagreb index

  

  
• Second modified Zagreb index

  

  
• General randić index

  

  

  Randić index for particular value at is

  
• Atom bond connectivity index

  

  
• Geometric arithmetic index

  

  
• Sum connectivity index

  

  
• Augmented Zagreb index

  

  

The numerical comparison of these indices can befound in Table 11 and their graphs are shown in Fig. 5.

Table 11.

Numerical comparison of degree-based indices of cyclohexane for n=1 to 10.

n				R(G)	ABC(G)	GA(G)	SCI(G)	AZI(G)
1	128	169	5.22	9.34	16.97	19.37	9.32	168.74
2	270	369	10.44	18.59	34.6	39.57	18.76	372.59
3	412	569	15.66	27.84	52.23	59.77	28.2	576.44
4	554	769	20.88	37.09	69.86	79.97	37.64	780.29
5	696	969	26.1	46.34	87.49	100.17	47.08	984.14
6	838	1169	31.32	55.59	105.12	120.37	56.52	1187.99
7	980	1369	36.54	64.84	122.75	140.57	65.96	1391.84
8	1122	1569	41.76	74.09	140.38	160.77	75.4	1595.69
9	1264	1769	46.98	83.34	158.01	180.97	84.84	1799.54
10	1406	1969	52.2	92.59	175.64	201.17	94.28	2003.39

Fig. 5.

Graphical comparison of degree-based TIs for cyclohexane. (a) Graphical Comparison of M₁(G),M₂(G), GA(G), AZI(G) for Cyclohexane. (b) Graphical Comparison of for Cyclohexane.

• Let G be a graph of the Cyclohexane network From Table 2 and Table 9, we have neighborhood degree-based topological indices which includes neighborhood Zagreb indices, neighborhood Randić index, the fourth atom bond connectivity index, the fifth geometric arithmetic index, the second ND index and sanskruti index are as follows:

• Third version of the Zagreb index

  

  
• Neighborhood second Zagreb index

  

  
• Neighborhood second modified Zagreb index

  

  
• Neighborhood general Randić index

  

  

  Randić index for particular value at is.

  
• The fourth atom bond connectivity index

  

  
• The fifth geometric arithmetic index

  

  
• The second ND index

  

  
• Sanskruti index

  

  

Table 12 shows the numerical behaviour of all given indices and their graphical structure is represented in Fig. 6.

Table 12.

Numerical comparison of neighborhood degree-based indices of cyclohexane for n=1 to 10.

n				NR(G)				S(G)
1	338	1234	0.48	3.23	6.91	20.61	5.13	1648.75
2	738	2907	0.88	6.23	18.21	42.1	10.74	4160.46
3	1138	4580	1.28	9.23	29.51	63.59	16.35	6672.17
4	1538	6253	1.68	12.23	40.81	85.08	21.96	9183.88
5	1938	7926	2.08	15.23	52.11	106.57	27.57	11695.59
6	2338	9599	2.48	18.23	63.41	128.06	33.18	14207.3
7	2738	11272	2.88	21.23	74.71	149.55	38.79	16719.01
8	3138	12945	3.28	24.23	86.01	171.04	44.4	19230.72
9	3538	14618	3.68	27.23	97.31	192.53	50.01	21742.43
10	3938	16291	4.08	30.23	108.61	214.02	55.62	24254.14

Fig. 6.

Graphical comparison of neighborhood degree-based TIs for cyclohexane. (a) Graphical Comparison of for Cyclohexane. (b) Graphical Comparison of for Cyclohexane.

Machine learning framework

Linear regression

Linear regression is a fundamental supervised machine-learning technique used to model and predict the relationship between a dependent variable and one or more independent variables by assuming a linear functional form as discussed in^33,34. In its simplest form, linear regression establishes a linear relationship between a single predictor (independent variable) and a response (dependent variable), making it an effective and interpretable tool for predictive analysis. The independent variable, also referred to as the predictor or explanatory variable, is assumed to be unaffected by other variables in the model (as mentioned in^35,36); however, variations in this variable directly influence the dependent variable. The dependent variable, commonly called the response or outcome variable, represents the quantity to be predicted or estimated. Linear regression aims to determine the optimal parameters that minimize the discrepancy between observed and predicted values, typically by employing the least squares estimation method.

Mathematically, the linear regression model is expressed as

The dependent variable is denoted by Y, the independent variable by X, the Y-intercept by b, and the independent variable’s coefficient by a.

Support vector regression

Support Vector Regression is a supervised machine learning technique applied to estimate the response of independent variables and a dependent variable³⁷. It does not attempt to have all the data point fall precisely, rather it tries to have a limited range of errors and prioritizes the most significant ones.

During SVR, a model is obtained after identifying a function whose error is within a constant threshold of requirement, which is in this case known as the error level, denoted as, ε. Data points that lie within this margin are thrown away and data points that lie out of the margin (which is termed as support vectors) are utilized in constructing the model. This causes SVR to be applicable to nonlinear and complicated data. Modeling of nonlinear relationships can also be done using kernel functions.

Mathematically, it is expressed as:

Ridge regression

Ridge regression is an advanced form of the linear regression which enhances the performance of a model when a data has a high number of variables or when the variables are very close to each other³⁸. It introduces a penalty term, which regulates the coefficients size.

In this approach, the independent variables are employed to predict dependent variable however, the model prevents the growth of the coefficients in an unacceptable way. This makes the model less prone to overfitting and makes it more stable.

Ridge regression is aimed at minimizing the error in predictions as well the values of the coefficients. Mathematically, it is written as:

Random forest regression

The random forest regression is a machine learning method of supervision through regression in which several decision trees give predictions³⁹. It eliminates using one model instead of utilizing a single model but instead a combination of several trees is utilized to enhance accuracy.

The decision trees are trained on separate parts of the data and the decision trees provide individual predictions. The last step is to get the average of all the predictions. This will minimize overfitting and maximize model reliability.

Random Forest enjoys popularity and finds application in many complex and nonlinear datasets.

Mathematically, it is expressed as:

Machine learning framework for cyclopentane

Prediction of chemical properties of cyclopentane through degree-based indices

Comparative analysis of Linear regression, Ridge regression, Support vector regression, and Random Forest models in predicting the physicochemical properties of Cyclopentane reveals that Linear Regression has the most optimal overall performance on most of the properties as indicated by the lowest values of MAE and RMSE and the highest scores of that are shown in table 13 and Figs. 7, 8, 9. In particular, Linear Regression has been seen to have excellent prediction capabilities of melting point, boiling point, density, flash point, log P, molar refractivity, and polarizability, where the value of the factors is typically higher than 0.90. Conversely, predictive error on several of the properties is significantly higher in SVR, although it has the largest value of the (0.9995) on this property, which indicates a good fit despite larger error values. Ridge and Random Forest models perform slightly better than linear model, and do not always perform better than the linear model. Also, the density of SVR is unsatisfactory, since in some instances it provides a negative value of implying to a poor fit. On the whole, these findings suggest that the majority of the physicochemical attributes of Cyclopentane are strongly correlated with the degree based indices, and thus the Linear Regression model is the most appropriate model in general, whereas the SVR would prove to be beneficial to individual nonlinear relationships, which include the prediction of molar weight.

Table 13.

The statistical values for best model of cyclopentane through degree-based indices.

Property	Model	MAE	RMSE
Melting point	Linear	1.5480	1.6148	0.9262
Boiling point	Linear	1.4561	1.7581	0.9866
Density	Linear	0.0020	0.0024	0.8600
Molar weight	SVR	0.8581	0.8813	0.9995
Flash point	Linear	0.3111	0.3800	0.8739
Log P	Linear	0.0238	0.0245	0.9033
Molar refractivity	Linear	0.2991	0.3599	0.9985
Polarizability	Linear	0.2038	0.2115	0.9970

Fig. 7.

MAE values of properties of cyclopentane through degree-based indices.

Fig. 8.

RMSE values of properties of cyclopentane through degree-based indices.

Fig. 9.

Values of properties of cyclopentane through degree-based indices.

Graphically representation of actual and predicted values of Cyclopentane on the basis of degree-based topological indices through all models are shown in Figs. 10, 11, 12, 13, 14, 15, 16, 17. In all figures, significant consistency is realized among the real values and model values on the real values of properties like boiling point, molar weight, flash point, log p, molar refractivity and polarizability in that the bars of predicted values almost coincide with the actual value and hence, high predictability. The Linear and Ridge regression models always demonstrate the most consistent correspondence with real values on all indices, which is the indicator of stable and consistent performance, and it can be argued that the correlation between the chosen indices and target attributes. The SVR and the Random Forest models are also effective only with slight variations, when the failure to predict the values particularly the melting point and molar weight works a bit but only by a slight margin, where slight underflattening and some overflattening are observed this effect is a bit more pronounced in the indices like AZI and GA. The variation in the melting point in models is relatively large when compared with other properties, but the differences are moderate and allowable. The precision of prediction of density and log P across all models and indices is very high making almost a perfect overlap. Notably, there is no significant index that can outperform the other rather, all indices show comparable predictive abilities, which underscores its effectiveness in capturing molecular structure information. The results obtained in degree-based indices support the strength and external applicability of the modeling strategy, as well as suggesting that most properties can be very accurately predicted using simpler models, and only small advantages are obtained with more complex models.

Fig. 10.

Actual vs predicted values of of cyclopentane through degree-based indices.

Fig. 11.

Actual vs predicted values of of cyclopentane through degree-based indices.

Fig. 12.

Actual vs predicted values of of cyclopentane through degree-based indices.

Fig. 13.

Actual vs predicted values of R of cyclopentane through degree-based indices.

Fig. 14.

Actual vs predicted values of ABC of cyclopentane through degree-based indices.

Fig. 15.

Actual vs predicted values of GAof cyclopentanethrough degree-based indices.

Fig. 16.

Actual vs predicted values of SCI of cyclopentane through degree-based indices.

Fig. 17.

Actual vs predicted values of AZI of cyclopentane through degree-based indices.

Prediction of chemical properties of cyclopentane through neighborhood degree-based indices

The relative performance of the neighborhood Degree-based indices as shown on MAE, RMSE and the value of plots as well as Table 14 and Figs. 18, 19, 20, proves the indicatively high predicting capability of all the models with the overall Linear Regression across most of the properties. The values of MAE and RMSE demonstrate that Linear Regression provides the lowest error values in melting point, boiling point, density, flash point, log p, and molar refractivity as well as polarizability and hence considerably accurate and stable predictions. Such performance is also demonstrated by a Ridge regression that is declared to be the best model about the molar weight with the lowest MAE (0.2142) and RMSE (0.2202). Conversely, SVR and Random Forest models show a comparatively greater error values of most of the properties especially boiling point and molar weight implying relatively lower precision predictions. The superiority of the models is further illustrated in the plot whose values are strictly near 1 and the values of the Regression being very near 1 which indicates an almost perfect between the predicted and the actual values.

Table 14.

The statistical values for best model of cyclopentane through neighborhood degree-based indices.

Property	Model	MAE	RMSE
Melting point	Linear	0.2064	0.2154	0.9996
Boiling point	Linear	0.1821	0.2198	0.9998
Density	Linear	0.0004	0.0005	0.9515
Molar weight	Ridge	0.2142	0.2202	0.9997
Flash point	Linear	0.0648	0.0792	0.9963
Log P	Linear	0.0093	0.0095	0.9757
Molar refractivity	Linear	0.1200	0.1444	0.9998
Polarizability	Linear	0.0680	0.0706	0.9998

Fig. 18.

MAE values of properties of cyclopentane through neighborhood degree-based indices.

Fig. 19.

RMSE values of properties of cyclopentane through neighborhood degree-based indices.

Fig. 20.

values of properties of cyclopentane through neighborhood degree-based indices.

In the vast majority of cases, the highest value of exceeding 0.999 of various properties, whereas in the cases of density and log p, there is a small but still high fit. In general, the findings show a good representation of the molecular structure by the neighborhood Degree-based indices, utilizing Linear Regression as the most successful and efficient model, and Ridge regression as having some marginal predictive in determining the molar weight.

The topological indices analyzed graphically by assessing neighborhood Degree-based indices reveals a very stable and corrective forecasting of the physicochemical properties of Cyclopentane using all machine learning models applied as can be seen in Figs. 21, 22, 23, 24, 25, 26, 27, 28, the predictive values of Linear regression, Ridge regression, SVR and Random Forest models greatly match the actual values of all properties such as melting point, boiling point, density, molar weight, flash point, log p, molar refractivity, as well as the polarizability, which means that the models perform well. Linear and Ridge regression models demonstrate almost perfect overlaps of actual values in each index, which points to their strength and the fact that the relationships between the depictions of neighborhoods and the target properties are largely linear. The SVR and the model (Random Forest) also give the correct predictions and have minimal differences in some instances, especially of melting point and molar weight, where slight underestimation or overestimation errors are observed; such changes are slight but have no significant effect on the performance. Nearly the same accuracy is predicted in properties like density, log P throughout all of models and indexes, with boiling point, molar refractivity, and polarizability also showing remarkably close agreement. Notably, there is no single index that particularly outsmarts the rest of the indices since all indices show an almost comparable predictive power, which means that all of them are equally useful in predicting the necessary molecular structural data. All in all, these findings prove the reliability, stability, and generalizability of the indices based on the degree of neighborhoods, but it also highlights that the less complex models such as Linear Regression can produce high levels of accuracy of predictions, and there is only slight comparison between the indices and models.

Fig. 21.

Actual vs predicted values of of cyclopentane through neighborhood degree-based indices.

Fig. 22.

Actual vs predicted values of of cyclopentane through neighborhood degree-based indices.

Fig. 23.

Actual vs predicted values of of cyclopentane through neighborhood degree-based indices.

Fig. 24.

Actual vs predicted values of NR of cyclopentane through neighborhood degree-based indices.

Fig. 25.

Actual vs predicted values of of cyclopentane htrough neighborhood degree-based indices.

Fig. 26.

Actual vs predicted values of of cyclopentane through neighborhood degree-based indices.

Fig. 27.

Actual vs predicted values of of cyclopentane through neighborhood degree-based indices.

Fig. 28.

Actual vs predicted values of S(G) of cyclopentane through neighborhood degree-based indices.

Machine learning framework for cyclohexane

Prediction of chemical properties of cyclohexane through degree-based indices

The results in terms of statistical table 15, along with the plots 29, 30, 31 of performance in terms of indices of degrees, supply a complete explanation of the higher accuracy and consistency of the Linear Regression model in predicting all the physicochemical properties of Cyclohexane, which is evidenced by all the values of the statistical table. Linear Regression has the lowest error values in both MAE and RMSE graphs of all the properties, including melting point and boiling point, density, molar weight, flash point, log p, molar refractivity, and polarizability showing that there is relative negligible deviation between actual and predicted values. Conversely, Ridge, the SVR and especially the Random Forest exhibit relatively higher errors, especially when it comes to molar weight, boiling point and the molar refractivity. This is even supported by the plot of in which Linear Regression also attains values of 1.0000to1.0009 indicating an almost perfect fit and therefore a very high degree of explanatory capacity compared to the other models, which attain somewhat lower values of the . These findings are quantified by the accompanying table, which indicates very low values of MAE (even as low as 0.0002 density), low values of RMSE (0.0003 density), and almost unit values of the R 2 of all the properties, all of which indicate that predictions are remarkably accurate and consistent. Affirmatively, density, and log P are anticipated with virtually perfect precision, whereas even more intricate characteristics as molar weight and refractivity pose extremely low error tolerances. In general, visual and tabular outcomes suggest that Degree-based indices are an effective way to reflect the information about the structure of Cyclohexane and that Linear Regression can be used to model such relationships with exceptional accuracy, significantly better than, seemingly, more sophisticated machine learning models and proving its efficiency and strength at the same time.

Table 15.

The statistical values for best model of cyclohexane through degree-based indices.

Property	Model	MAE	RMSE
Melting point	Linear	0.1509	0.1615	0.9995
Boiling point	Linear	0.1311	0.1583	0.9997
Density	Linear	0.0002	0.0003	0.9999
Molar weight	Linear	0.1714	0.1762	0.9999
Flash point	Linear	0.0519	0.0633	0.9993
Log P	Linear	0.0119	0.0124	0.9991
Molar refractivity	Linear	0.0900	0.1083	0.9997
Polarizability	Linear	0.0544	0.0565	0.9998

Fig. 29.

MAE values of properties of cyclohexane through degree-based indices.

Fig. 30.

RMSE values of properties of cyclohexane through degree-based indices.

Fig. 31.

values of properties of cyclohexane through degree-based indices.

The graphical findings 32, 33, 34, 35, 36, 37, 38, 39 of the Degree-based topological indices indicate a very reliable and precise predictive trend of the physicochemical characteristic of Cyclohexane in all the machine learning models applied in them and especially between the actual and the predicted properties. The values of the Linear, Ridge, SVR, and Random Forest model are almost equal to actual values on all properties, such as melting point, boiling point, density, molar weight, flash point, log P, molar refractivity, and polarizability, which means that there is little deviation, and thus predictive properties are high. The discrepancies between models are quite minor, although Linear and Ridge regression models tend to fit best the reality, whereas SVR and Random Forest exhibit minor variations, particularly in such features as boiling point and molar refractivity. Density and log P are determined with virtually perfect accuracy across all indices, and indicate a great dependence on structural descriptors, but with molar weight and flash point both great agreement is found with only small error. Besides, the fact that the results are consistent when different indices are involved indicates that all the Degree-based descriptors do a good job in the determination of the structural properties of cyclohexane, meaning that the results are robust and stable despite the index of choice. Generally, these plots indicate that the indices that are used with the relative simplicity of the regression models are enough to produce extremely accurate modeling and that Linear regression is somewhat better in achieving the proximity of the actual value but also that more advanced models would not offer any substantial further advantage.

Fig. 32.

Actual vs predicted values of of cyclohexane through degree-based indices.

Fig. 33.

Actual vs predicted values of of cyclohexane through degree-based indices.

Fig. 34.

Actual vs predicted values of of cyclohexane through degree-based indices.

Fig. 35.

Actual vs predicted values of R of cyclohexane through degree-based indices.

Fig. 36.

Actual vs predicted values of ABC of cyclohexane through degree-based indices.

Fig. 37.

Actual vs predicted values of GA of cyclohexane through degree-based indices.

Fig. 38.

Actual vs predicted values of SCI of cyclohexane through degree-based indices.

Fig. 39.

Actual vs predicted values of AZI of cyclohexane through degree-based indices.

Prediction of chemical properties of cyclohexane through neighborhood degree-based indices

The statistical comparison of the model behaviours based on the MAE, RMSE and indicator, shown in table 16 and Figs. 40, 41, 42. The Linear Regression model gives the lowest values of the errors in most of the physicochemical properties, such as melting point (MAE = 0.2064, RMSE = 0.2154), boiling point (MAE = 0.1821, RMSE = 0.2198), flash point, log P, molar refractivity and polarizability and it also demonstrates exceptionally high values of R2 near to unity, which means that it is an excellent predictor and has a strong correlation to the real The model also predicts density with the lowest error (MAE = 0.0004, RMSE = 0.0005) although its R square value (= 0.9515) is a bit less than that of other properties indicating some uncertainty with which it explains its distribution. Ridge Regression compares well and even better than Linear Regression in molar weight prediction, giving the best combination of error (MAE = 0.2142, RMSE = 0.2202) and almost perfect R2 (= 0.9997). Comparatively, SVR depicts moderate values of error between properties, whereas Random Forest depicts the greatest values of MAE and RMSE, specifically in boiling point and molar weight indicating a worse performance although it still captures high values of the values of . All in all, the findings indicate that Degree-based indices of the neighborhood are the most effective descriptors of cyclohexane, and linear-based models, specifically Linear Regression offer the best predictive accuracy, but does not imply that increasing the model complexity, in this case, is a successful way to achieve the best results.

Table 16.

The statistical values for best model of cyclohexane through neighborhood degree-based indices.

Property	Model	MAE	RMSE
Melting point	Linear	0.2064	0.2154	0.9996
Boiling point	Linear	0.1821	0.2198	0.9998
Density	Linear	0.0004	0.0005	0.9515
Molar weight	Ridge	0.2142	0.2202	0.9997
Flash point	Linear	0.0648	0.0792	0.9963
Log P	Linear	0.0093	0.0095	0.9757
Molar refractivity	Linear	0.1200	0.1444	0.9998
Polarizability	Linear	0.0680	0.0706	0.9998

Fig. 40.

MAE values of properties of cyclohexane through neighborhood degree-based indices.

Fig. 41.

RMSE values of properties of cyclohexane through neighborhood degree-based indices.

Fig. 42.

values of properties of cyclohexane through neighborhood degree-based indices.

A comparative graphical analysis of Cyclohexane on the basis of neighborhood Degree-based indices reveals that there are very high predictive accuracy and consistency across all physicochemical properties and regression formulas that graphs are shown in 43, 44, 45, 46, 47, 48, 49, 50. In terms of each index, the values extrapolated using Linear, Ridge, SVR, and Random Forest models are close to the actual values in the experiment, which means that there is minimal deviation and high stability of the models. Descriptors like melting point, boiling point, molar weight, and flash point are also very robust with almost similar predicted values across all models, with only small differences being observed. In a corresponding fashion, the similar variables density and log P, which are generally smaller in magnitude, are forecasted with exceptionally accuracy in all the indices. Linear and Ridge Regression have the best fit with the actual values of all the models, even though both SVR and Random Forest are also close in overall fit, only sometimes with very slight under or over-fitting especially with larger magnitude values as in boiling point. The predictions have no significant variation or unsteadiness in any indices, indicating that these neighborhood Degree-based descriptors are capable of fitting well the structural information of cyclohexane. In general, the findings have functioned to substantiate that these indices, no matter how they may be formulated, are extremely dependable in modeling and forecasting physicochemical properties and that more simple linear-based models can be utilized to obtain near-perfect predictive performance without having to utilize more complicated nonlinear models.

Fig. 43.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 44.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 45.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 46.

Actual vs predicted values of NR of cyclohexane through neighborhood degree-based indices.

Fig. 47.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 48.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 49.

Actual vs predicted values of of cyclohexane through neighborhood degree-based indices.

Fig. 50.

Actual vs predicted values of S(G) of cyclohexane through neighborhood degree-based indices.

Comparison and discussion

The topological indices applied in this paper to examine the molecular structures of Cyclopentane and Cyclohexane are of the following two categories: degree based indices and the neighborhood degree based indices. There are the Degree-based indices the First Zagreb index and Second Zagreb index, the Modified Zagreb index, Randi c index, the Atom Bond Connectivity index, Geometric Arithmetic index, the Sum Connectivity Index, and the Augmented Zagreb index. The major indices used are used to characterize the simplicity of connectivity of atoms of a graph of molecules using vertex degrees.

In an attempt to assess the validity of these indices in forecasting physicochemical properties, various machine learning algorithms have been used, namely, Linear Regression, Ridge Regression, Support Vector Regression (SVR), and Random Forest. Linear Regression is indicated to be the best in most of the properties, with a minimum error values (MAE and RMSE) and a high value of the variance, which shows it to exhibit high predictive accuracy. Ridge Regression is also effective and especially effective on some properties which include molar weight. Conversely, errors in the case of SVR and Random Forest models are relatively higher and errors are not always consistent, yet in certain instances, both allow reasonable predictions.

On the whole, the research results explain that the two kinds of indexes both work well in modelling the molecular properties, but the neighbourhood Degree-based indices are more predictive because they are capable of describing more structural detail. Also, its results point out the fact that basic linear models are adequate and they tend to be much more reliable than more complex models to work with this kind of data, so the results of using simple linear models are effective and correct tools of chemical property prediction.

Conclusion

The analysis performed in this study is a thorough graph-theoretical and machine-learning-based analysis of the molecular graph of Cyclopentane and Cyclohexane. Both Degree-based topological indices, and neighborhood Degree-based topological indices were both calculated and turn out to be in closed form, with obvious linear behaviour with the repetitive units. The comparative outcomes expose that Cyclohexane tends to achieve a bigger index than those of Cyclopentane due to its comparatively larger and more networked grouping whereas indexes of neighborhoods offer a more detailed and informative portrayal of museum structure in comparison with easy tools of description of degree. The machine learning analysis also displays that these topological indices are very much effective in the predicting of important physics chemical characteristics, which comprise melting point, boiling point, density, molar weight, flash point, log P, molar refractivity, and polarizability. Linear Regression indicates the best and most consistent and accurate result of the tested models, although Ridge Regression involves minor improvements in molar weight with certain neighborhood-related tests, and SVR or Random Forest contributes to no significant overall benefit. Hence, the paper finds that simple linear models along with topological descriptors (particularly the indices based on neighborhood degree) are an effective, trustworthy and computationally economical representation of the leaves of QSPR analysis and molecular property prediction of polymer-related structures like Cyclopentane and Cyclohexane.

Author contributions

All the authors have equally contributed to this manuscript. However the main contribution of each author is as follows. Muhammad Irfan (Conceptualization, Investigation, Methodology, Project administration, Writing – original draft, Writing – review, Supervision), Nabeela Bashir (Conceptualization, Investigation, Software, Writing – original draft, Visualization), M. Umair Shahzad (Formal analysis, Project administration, Software, Resources). Kamal M.Othman (Formal analysis, Funding acquisition, Project administration, Software, Resources). Abdulfattah Noorwalli (Funding acquisition, Software, Supervision, Project administration). AbdulGuddoos S. A. Gaid (Formal analysis, Project administration, Software, Resources). Esam Y.O. Zafar (Formal analysis, Project administration, Software, Resources). All authors agree to be accountable for the content and conclusions of the article.

Funding

No Funding.

Data availability

All data generated or analysed during this study are included in this published article [and its supplementary information files]

Declarations

Competing interests

The authors declare no competing interests.