Leveraging topological indices and machine learning for advanced prediction of antidepressant drug properties

Abstract

This study investigates the efficacy of eight multiplicative degree-based and three classical degree-based topological indices in Quantitative Structure-Property Relationship (QSPR) models for predicting critical physicochemical properties of 38 antidepressant drugs. Molecular structures of compounds including Bupropion, Amitriptyline, and Fluoxetine were translated into numerical descriptors using indices such as Multiplicative Sum Zagreb, Multiplicative Sombor, and the First Zagreb index. These descriptors were integrated with machine learning algorithms: Random Forest, XGBoost, and linear regression to forecast boiling points, melting points, critical temperature, critical volume, and molar refractivity. Results revealed that the XGBoost algorithm significantly outperformed other methods, achieving superior predictive accuracy with the lowest error metrics (e.g., for boiling point: MAE = 8.60, RMSE = 12.40, ). Among the topological indices, the First Zagreb index () emerged as the most robust descriptor, demonstrating the strongest correlations with key properties (e.g., with critical volume, with molar refractivity). Linear regression models further confirmed the significance of and other indices, with high statistical significance () in most cases. This interdisciplinary approach demonstrates the potent synergy of graph-theoretic indices and advanced machine learning in pharmaceutical research, offering a powerful strategy to accelerate drug discovery and optimize the design of novel therapeutic agents.

Introduction

Chemical graph theory provides a powerful framework for representing molecular structures as mathematical graphs, where atoms and bonds are denoted as vertices and edges, respectively ^1–3. In this context, a molecular structure is abstracted as a mathematical graph G(V, E), where a set of vertices V represents atoms and a set of edges E represents chemical bonds between them. The degree of a vertex , a fundamental concept, is the number of edges incident to it, corresponding to the valency of an atom. A topological index is then a numerical descriptor, or a graph invariant, calculated from this graph G to encode its structural information such as molecular size, shape, branching, and cyclicity. The foundational work of Wiener, who linked molecular structure to alkane boiling points ⁴, ignited the field of Quantitative Structure-Property Relationship (QSPR) modeling. In QSPR, TIs serve as critical predictors for correlating molecular structure with physicochemical and biological properties, thereby offering a computational shortcut for expensive experimental measurements.

The evolution of TIs has been marked by continuous innovation to enhance their predictive power. The Randic index, introduced to quantify molecular branching ⁵, and the Zagreb indices, developed by Gutman and Trinajstic ⁶, are among the most influential classical descriptors. Recent advancements have focused on multiplicative variants of these indices, such as the multiplicative Zagreb indices proposed by Eliasi et al. ⁵, which have demonstrated improved adaptability and accuracy. This has been followed by the development of other multiplicative indices, including the Hyper Zagreb index for thermal predictions ⁷ and the Multiplicative Sombor index to expand the descriptor toolbox ⁸.

The topological indices considered in this work, summarized in Table 1, were selected to provide a comprehensive and modern characterization of molecular structure for QSPR modeling. This set includes both classical workhorses of chemical graph theory and their contemporary multiplicative variants. We examine the foundational First and Second Zagreb indices (, ) and the First Hyper Zagreb index (), which are based on the sum of vertex degrees and their squares. Alongside these, we compute their multiplicative analogues: the Multiplicative Sum Zagreb (), Multiplicative Zagreb (), and Multiplicative First Hyper Zagreb () indices which operate on the product of terms over edges, often yielding greater sensitivity to structural nuances. To further expand the descriptor space, we also include the generalized versions for (, ), the recently introduced Multiplicative Sombor index () based on Euclidean distances, and the multiplicative form of the seminal Randic index (MR(G)). The diverse portfolio of indices captures a wide spectrum of topological information, from basic connectivity and branching patterns to more complex electronic and steric influences, providing a robust foundation for predictive models ^9–12.

Table 1.

Definitions and pioneers of the degree-based topological indices used in this study.

Names of indices	Definitions of indices	Pioneer’s names
Multiplicative Sum Zagreb()		Eliasi et al.5
Multiplicative First Hyper Zagreb()		Kulli7
Multiplicative Zagreb()		Gutman (2011)
Multiplicative Second Hyper Zagreb()		Kulli7
Generalized version of Multiplicative Sum ZI ()()		Sun and Du (2024)
Generalized version of Multiplicative ZI ()()		Alfuraidan et al. (2020)
Multiplicative Sombor()		Liu8
Multiplicative Randic(MR(G))		Gutman et al. (2016)
First Zagreb index()		Gutman and Trinajstic6
Second Zagreb index()		Gutman and Trinajstic6
First Hyper Zagreb index()		Shirdel et al. (2013)

Depression is a multifaceted neuropsychiatric disorder that is often treated with a wide spectrum of pharmacological agents, each targeting different neurotransmitter systems. The drugs studied in this work including Bupropion, Amitriptyline, Effexor (Venlafaxine), Fluoxetine, Nefazodone, Paxil (Paroxetine), Sertraline, and Trazodone primarily function by modulating monoamine levels such as serotonin, norepinephrine, and dopamine in the brain ^13–15. Newer adjuncts like Gabapentin, Buspirone, and Risperidone have been employed to augment traditional antidepressants, especially in treatment-resistant cases ¹⁶. Compounds like L-tyrosine, L-theanine, Folate, and Dehydroepiandrosterone are emerging as nutraceuticals or hormonal adjuncts with mood-stabilizing or neuroprotective effects ^17,18. Drugs like Amphetamine and Pregnenolone, though not primarily antidepressants, are under research for their potential in enhancing motivation and cognitive function in depressive disorders ¹⁹. This pharmacological diversity reflects the complexity of depression and underscores the importance of predictive tools, such as QSPR models, for better understanding drug behavior and optimizing therapeutic outcomes.

While Quantitative Structure-Property Relationship (QSPR) studies are well-established, and machine learning (ML) is increasingly applied in cheminformatics, the specific integration of a comprehensive set of modern multiplicative topological indices with state-of-the-art ML algorithms for the targeted prediction of antidepressant drug properties represents a significant and underexplored frontier. ML techniques like Random Forest and XGBoost excel at recognizing complex, non-linear patterns in high-dimensional data, offering a potent complement to traditional linear regression ^20,21. Classical degree-based indices like the Zagreb and Randic indices have a long history in QSPR; however, their multiplicative variants offer enhanced sensitivity to nuanced structural patterns due to their product-based formulations, which may better capture complex molecular interactions. This study fills a critical gap by systematically evaluating the predictive power of eight multiplicative and three classical indices including recently proposed descriptors like the Multiplicative Sombor index when coupled with powerful non-linear ML models (Random Forest and XGBoost) specifically for a structurally diverse set of 38 antidepressant compounds. Our work thus provides a novel, interdisciplinary framework that uniquely leverages the synergy between advanced graph-theoretic descriptors and modern ML to optimize the predictive modeling of critical physicochemical properties in antidepressant research.

This study aims to bridge this gap by conducting a rigorous QSPR analysis on 38 depression-related compounds. We compute eight multiplicative and three classical degree-based TIs to serve as molecular descriptors. These indices are then employed within both linear regression and automated ML models (Random Forest and XGBoost) to predict key physicochemical properties. Our work demonstrates that this interdisciplinary methodology, marrying graph-theoretic concepts with advanced ML, provides a highly accurate and efficient framework for property prediction, ultimately contributing to the accelerated discovery and optimization of novel antidepressant therapeutics.

Computation of topological indices

This section details the computation of a suite of multiplicative and classical degree-based topological indices for a set of 38 antidepressant drugs. The analyzed compounds include widely used therapeutics such as Bupropion, Amitriptyline, Fluoxetine, and Sertraline, as well as nutraceutical adjuncts like L-tyrosine and L-theanine, providing a structurally diverse dataset for our QSPR study. The complete list of drugs considered in this study is: Bupropion, Amitriptyline, Effexor, Fluoxetine, Nefazodon, Paxil, Sertraline, Trazodone, Nomifensine, Valproate, Risperidone, Buspirone, Gabapentin, Dopamine, L-tyrosine, L-theanine, Mianserin, Folate, Meprobamate, Amphetamine, Dehydroepiandrosterone, Pregnenolone, Citalopram, Paroxetine, Duloxetine, Venlafaxine, Desvenlafaxine, Levomilnacipran, Clomipramine, Nortriptyline, Mirtazapine, Vilazodone, Tranylcypromine, Esketamine, Agomelatine, Moclobemide, Clorgiline, and Tianeptine. The molecular structures of these drugs are depicted in Figs. 1, 2, and 3 respectively.

Fig. 1.

Molecular structures of antidepressant drugs (Bupropion to L-theanine).

Fig. 2.

Molecular structures of antidepressant drugs (Mianserin to Clomipramine).

Fig. 3.

Molecular structures of antidepressant drugs (Nortriptyline to Tianeptine).

The computation of topological indices first requires a systematic analysis of the molecular graph. For each compound, we identified its edge partition based on the degrees of the adjacent vertices for every edge in the graph G. This partition serves as the foundational data from which all indices are derived, as their formulas are applied by grouping edges of the same type. To illustrate this computational process, we consider Amitriptyline as a detailed example. Its molecular graph G consists of 21 vertices and 22 edges. The edge partition, categorized by the degree pairs of connected vertices, is as follows: 2 vertices of type (1, 3), 9 vertices of type (2, 2), 8 vertices of type (2, 3), and 4 vertices of type (3, 3).

Using this partition, the values for the topological indices defined in Table 1 are calculated by applying their respective formulas. For instance, the Multiplicative Sum Zagreb index is computed as the product of the sum of vertex degrees for each edge:

The calculations for the remaining indices for Amitriptyline, including the Multiplicative First Hyper Zagreb (), Multiplicative Zagreb (), and Multiplicative Randic (MR(G)), proceed analogously. The computed values for all indices for Amitriptyline are:

To ensure accuracy and efficiency in calculating indices for the entire set of 38 drugs, we developed a dedicated Python-based computational tool. This program automates the process of parsing molecular structures, determining edge partitions, and computing all defined indices. The resulting edge partition for every drug is tabulated in Table 2, providing a transparent record for verification and reproducibility. The final computed values of all eleven topological indices for all 38 antidepressant drugs are comprehensively presented in Table 3. These numerical descriptors serve as the feature set for the subsequent machine learning and QSPR modeling analyses detailed in the following sections.

Table 2.

Edge partition based on the degrees of adjacent vertices for the molecular graphs of the 38 antidepressant drugs.

Drugs	Edge partition
	(1,2)	(1,3)	(1,4)	(2,2)	(2,3)	(2,4)	(3,3)	(3,4)
Bupropion	0	3	3	3	3	1	3	0
Amitriptyline	0	2	0	9	8	0	4	0
Effexor	1	2	1	6	7	2	1	1
Fluoxetine	1	0	3	8	9	0	1	1
Nefazodon	1	2	0	12	17	0	4	0
Paxil	0	1	0	9	14	0	3	0
Sertraline	1	2	0	5	9	0	5	0
Trazodone	0	2	0	9	14	0	4	0
Nomifensine	0	2	0	6	8	0	4	0
Valproate	2	2	0	2	2	0	1	0
Risperidone	0	3	0	8	16	0	7	0
Buspirone	0	2	0	12	10	4	3	0
Gabapentin	1	2	0	4	1	4	0	0
Dopamine	1	2	0	2	5	0	1	0
L-tyrosine	0	4	0	2	6	0	1	0
L-theanine	1	4	0	2	3	0	1	0
Mianserin	0	1	0	7	10	0	5	0
Folate	0	7	0	5	18	0	4	0
Meprobamate	1	5	0	1	5	0	1	0
Amphetamine	0	2	0	4	4	0	0	0
Dehydroepiandrosterone	0	2	2	4	8	2	2	4
Pregnenolone	0	3	2	4	8	2	3	4
Citalopram	0	4	0	6	10	2	1	2
Paroxetine	0	1	0	9	15	0	2	0
Duloxetine	1	0	0	10	10	0	3	1
Venlafaxine	0	2	1	8	6	3	0	0
Desvenlafaxine	0	4	1	2	6	3	1	1
Levomilnacipran	0	4	0	5	8	0	3	0
Clomipramine	0	3	0	7	10	0	4	0
Nortriptyline	1	0	0	7	5	0	2	0
Mirtazapine	0	1	0	5	8	0	3	0
Vilazodone	0	3	0	6	6	0	3	0
Tranylcypromine	0	0	0	7	4	0	1	0
Esketamine	0	3	0	6	4	0	4	0
Agomelatine	0	1	0	8	6	0	4	0
Moclobemide	1	2	0	4	11	0	1	0
Clorgiline	1	1	0	4	5	0	3	0
Tianeptine	0	2	0	8	8	0	1	0

Table 3.

Computed values of the eleven topological indices for all 38 antidepressant drug structures.

Drugs	Topological indices
								MR(G)
Bupropion									78	86	390
Amitriptyline									108	126	520
Effexor									100	115	494
Fluoxetine									108	121	522
Nefazodon									168	194	802
Paxil									128	150	618
Sertraline									106	127	526
Trazodone									138	162	670
Nomifensine									96	114	472
Valproate									38	39	168
Risperidone									166	200	828
Buspirone									96	114	472
Gabapentin									56	62	274
Dopamine									50	54	234
L-tyrosine									60	64	282
L-theanine									48	48	216
Mianserin									112	136	558
Folate									162	185	786
Meprobamate									58	60	266
Amphetamine									44	46	196
Dehydroepiandrosterone									126	160	686
Pregnenolone									136	172	738
Citalopram									122	145	616
Paroxetine									127	147	607
Duloxetine									112	130	534
Venlafaxine									93	102	443
Desvenlafaxine									90	105	464
Levomilnacipran									94	107	452
Clomipramine									114	133	554
Nortriptyline									69	79	325
Mirtazapine									82	98	404
Vilazodone									84	96	402
Tranylcypromine									54	61	248
Esketamine									80	93	388
Agomelatine									90	107	438
Moclobemide									88	99	416
Clorgiline									66	78	322
Tianeptine									86	95	396

Predictive modeling using topological indices

The computed topological indices for the 38 antidepressant drugs serve as the feature set for constructing predictive Quantitative Structure-Property Relationship (QSPR) models. To comprehensively evaluate the predictive power of these indices, we employ a multi-faceted computational approach utilizing three distinct regression techniques: Linear Regression, Random Forest, and Extreme Gradient Boosting (XGBoost). Each method offers unique advantages. Linear Regression provides a transparent, interpretable baseline model, establishing fundamental linear relationships between the indices and the physicochemical properties. Random Forest, an ensemble learning method, enhances predictive accuracy by building a multitude of decision trees and aggregating their results, effectively capturing non-linear patterns and interactions within the data. Finally, XGBoost, a sophisticated gradient-boosting framework, is employed to optimize predictive performance further; it sequentially builds models that correct the errors of previous ones, often yielding state-of-the-art results in machine learning competitions and scientific applications. This progression from a simple linear model to complex, high-performance algorithms allows for a robust assessment of how well topological indices can predict drug properties under varying levels of model complexity.

Linear regression model

We begin our predictive analysis with Linear Regression, a fundamental and interpretable statistical method that models the linear relationship between a dependent variable and one or more independent variables. In the context of our QSPR study, the goal is to establish a direct linear relationship between each topological index (TI), serving as the independent variable, and the physicochemical properties (P) of the antidepressant drugs, which are the dependent variables.

The general form of the simple linear regression model used for each index-property pair is given by:

1

where:

• P represents a physicochemical property (Boiling Point, Melting Point, Critical Temperature, Critical Volume, or Molar Refractivity).

• a is the y-intercept of the regression line.

• b is the regression coefficient (slope), indicating the change in the property for a unit change in the topological index.

• TI represents the value of a topological index.

The experimental values for the five physicochemical properties (boiling points (BP), melting points (MW), critical temprature (CT), critical volume (CV) and molar refractivity (MR)) were sourced from ChemDraw and PubChem and are compiled in Table 4. Using the SPSS statistical software, we generated a distinct linear regression model for each combination of the eleven topological indices and the five properties, resulting in a comprehensive set of 55 models.

Table 4.

Experimentally determined physicochemical properties for the 38 antidepressant drugs.

Drugs Names	Physiochemical properties
	Boiling Point	Melting Point	Critical Temp.	Critical Vol.	MR
Bupropion	666.50	394.64	778.39	728.50	68.70
Amitriptyline	751.01	457.55	856.98	898.50	93.40
Effexor	766.61	451.59	812.60	859.50	83.18
Fluoxetine	713.63	410.29	803.42	861.50	81.04
Nefazodon	1092.30	812.28	1072.62	1308.50	132.29
Paxil	853.05	600.10	907.55	888.50	90.51
Sertraline	788.23	493.93	872.67	852.50	85.99
Trazodone	919.92	716.19	994.91	984.50	106.26
Nomifensine	735.32	481.13	874.37	719.50	77.56
Valproate	457.79	252.95	620.30	492.50	–
Risperidone	1068.52	822.57	1034.63	1144.50	114.21
Buspirone	1045.74	850.24	1065.22	1117.50	108.44
Gabapentin	643.35	465.83	784.09	523.50	45.22
Dopamine	643.09	512.54	800.17	436.50	42.63
L-tyrosine	730.42	557.69	831.85	494.50	46.62
L-theanine	702.27	563.22	872.17	502.50	42.53
Mianserin	713.52	478.58	865.92	786.50	85.92
Folate	1500.13	1358.86	1399.64	1186.50	–
Meprobamate	663.79	444.29	755.19	638.50	53.73
Amphetamine	504.29	285.37	700.68	454.50	43.69
Dehydroepiandrosterone	837.91	539.19	850.34	893.50	84.02
Pregnenolone	865.05	539.20	849.27	1003.50	93.11
Citalopram	873.22	571.52	890.36	977.50	94.41
Paroxetine	833.56	893.48	911.40	885.50	92.57
Duloxetine	805.31	536.28	906.24	864.50	89.70
Venlafaxine	674.87	405.77	809.54	846.50	82.01
Desvenlafaxine	738.02	504.79	817.09	701.50	68.50
Levomilnacipran	750.48	516.93	857.50	767.50	75.32
Clomipramine	754.45	479.31	861.87	960.50	97.93
Nortriptyline	597.85	360.07	803.75	612.50	61.17
Mirtazapine	691.85	566.66	853.81	588.50	59.84
Vilazodone	694.20	467.86	840.85	693.50	71.85
Tranylcypromine	483.20	279.51	734.59	455.50	47.78
Esketamine	682.82	440.65	847.57	650.50	70.88
Agomelatine	719.40	537.19	871.67	656.50	68.51
Moclobemide	754.38	548.22	928.78	733.50	74.24
Clorgiline	594.50	379.97	810.89	549.50	56.24
Tianeptine	713.06	437.54	833.21	734.50	69.75

The derived regression equations for all index property pairs are listed below.

1. Multiplicative Sum Zagreb

   
2. Multiplicative First Hyper Zagreb

   
3. Multiplicative Zagreb

   
4. Multiplicative Second Hyper Zagreb

   
5. Generalized Version of Multiplicative Sum Zagreb

   
6. Generalized Version of Multiplicative Zagreb

   
7. Multiplicative Sombor

   
8. Multiplicative Randic

   
9. First Zagreb index

   
10. Second Zagreb index

    
11. First Hyper Zagreb index

    

The model performance and statistical significance were rigorously evaluated. A detailed statistical overview, including standard error of the estimate (SEE), F-statistic, t-statistic, and p-values for each model, is provided in Tables 7, 8, 9, 10, and 11. These results confirm that the majority of the models are statistically significant (). Notably, the First Zagreb index () consistently emerged as the most powerful predictor across all properties, exhibiting the strongest linear correlations (e.g., with Critical Volume, with Molar Refractivity).

Table 7.

Statistical overview of the simple linear regression QSPR models for boiling point (BP).

Topological indices	r	r2	SEE	F	p	t	Indicator
	0.4167	0.1737	171.0836	7.56513		2.75048	Significant
	0.3160	0.0998	178.5622	3.99235		1.99809	Not significant
	0.4288	0.1838	170.0266	8.10849		2.84754	Significant
	0.3921	0.1538	173.1313	6.54069		2.55748	Significant
	0.2988	0.0893	179.6065	3.52862		1.87846	Not significant
	0.3734	0.1395	174.5869	5.83432		2.41543	Significant
	0.5786	0.3348	153.5027	18.1158		4.25626	Significant
	0.3593	0.1291	175.6339	5.33701		2.31020	Significant
	0.8236	0.6783	106.7521	75.8931		8.71166	Significant
	0.8009	0.6414	112.7050	64.3851		8.02403	Significant
	0.8074	0.6519	111.0323	67.4324		8.21173	Significant

Table 8.

Statistical overview of the simple linear regression QSPR models for melting point (MP).

Topological indices	r	r2	SEE	F	p	t	Indicator
	0.3466	0.1202	190.3268	4.91675		2.21737	Significant
	0.2485	0.0618	196.5431	2.36946		1.53931	Not significant
	0.3541	0.1254	189.7629	5.16031		2.27163	Significant
	0.3163	0.1001	192.4895	4.00252		2.00063	Not significant
	0.2325	0.0541	197.3469	2.05751		1.43440	Not significant
	0.2986	0.0892	193.6481	3.52526		1.87757	Not significant
	0.5130	0.2631	174.1793	12.8549		3.58537	Significant
	0.2900	0.0841	194.1866	3.30634		1.81833	Not significant
	0.6890	0.4747	147.0673	32.5281		5.70334	Significant
	0.6611	0.4370	152.2443	27.9468		5.28647	Significant
	0.6624	0.4388	152.0085	28.1454		5.30522	Significant

Table 9.

Statistical overview of the simple linear regression QSPR models for critical temperature (CT).

Topological indices	r	R2	SEE	F	p	t	Indicator
	0.1085	0.0118	345.0501	0.429215		0.655145	Not significant
	0.0793	0.0063	346.0065	0.228106		0.477604	Not significant
	0.1023	0.0105	345.2783	0.381082		0.617319	Not significant
	0.0925	0.0086	345.6125	0.310767		0.557465	Not significant
	0.0749	0.0056	346.1260	0.203100		0.450666	Not significant
	0.0892	0.0080	345.7167	0.288873		0.537469	Not significant
	0.1607	0.0258	342.5919	0.953873		0.976664	Not significant
	0.1865	0.0348	341.0092	1.29769		1.13916	Not significant
	0.2242	0.0503	338.2622		1.38056	Not significant
	0.1999	0.0400	340.0952	1.49844		1.22411	Not significant
	0.1937	0.0375	340.5288	1.40301		1.18449	Not significant

Table 10.

Statistical overview of the simple linear regression QSPR models for critical volume (CV).

Topological indices	r	r2	SEE	F	p	t	Indicator
	0.4930	0.2431	192.5764	11.5622		3.40032	Significant
	0.4248	0.1805	200.3838	7.92815		2.81570	Significant
	0.5136	0.2638	189.9286	12.8976		3.59132	Significant
	0.4965	0.2465	192.1418	11.7776		3.43186	Significant
	0.4107	0.1687	201.8232	7.30383		2.70256	Significant
	0.4827	0.2330	193.8533	10.9377		3.30722	Significant
	0.5733	0.3287	181.3620	17.6260		4.19834	Significant
	0.3544	0.1256	206.9849	5.17096		2.27397	Significant
	0.9317	0.8681	80.3816	236.995		15.3946	Significant
	0.9179	0.8425	87.8328	192.641		13.8795	Significant
	0.9201	0.8466	86.6989	198.661		14.0947	Significant

Table 11.

Statistical overview of the simple linear regression QSPR models for Molar Refractivity (MR).

Topological indices	r	r2	SEE	F	p	t	Indicator
	0.4961	0.2461	19.1565	11.1014		3.33187	Significant
	0.4536	0.2058	19.6629	8.80825		2.96787	Significant
	0.5364	0.2877	18.6205	13.7357		3.70617	Significant
	0.5312	0.2821	18.6938	13.3619		3.65539	Significant
	0.4420	0.1954	19.7908	8.25672		2.87345	Significant
	0.5177	0.2680	18.8762	12.4508		3.52857	Significant
	0.5300	0.2808	18.7104	13.2780		3.64390	Significant
	0.5350	0.2862	18.6406	13.6325		3.69222	Significant
	0.9382	0.8802	7.6376	249.732		15.8029	Significant
	0.9237	0.8532	8.4522	197.675		14.0597	Significant
	0.9164	0.8398	8.8314	178.211		13.3496	Significant

A detailed comparison between the actual experimental values and the values predicted by the linear regression models is systematically presented in Tables 5 and 6. These predictions were derived using the most statistically significant regression model namely, the one based on the First Zagreb index which consistently demonstrated the strongest linear correlations across all physicochemical properties. The scatter plots in Fig. 4 visually reinforce these relationships by illustrating the correlation between the actual property values and the topological indices. While these linear models establish a robust and interpretable baseline, they also highlight the inherent limitations of linear approximations, thereby paving the way for the application of more sophisticated, non-linear machine learning techniques to further enhance predictive accuracy.

Table 5.

Comparison of actual values and linear regression predicted values for boiling point (BP), melting point (MP), and critical temperature (CT).

Drugs	Properties
	BP		MP		CT
Structures Name	Actual	Predicted	Actual	Predicted	Actual	Predicted
Bupropion	666.50	683.84	394.64	465.12	778.39	880.46
Amitriptyline	751.01	819.98	457.55	587.91	856.98	948.83
Effexor	766.61	783.68	451.59	555.16	812.60	930.60
Fluoxetine	713.63	819.98	410.29	587.91	803.42	948.83
Nefazodon	1092.30	1092.26	812.28	833.49	1072.62	1085.57
Paxil	853.05	910.74	600.10	669.77	907.55	994.41
Sertraline	788.23	810.91	493.93	579.72	872.67	944.27
Trazodone	919.92	956.12	716.19	710.70	994.91	1017.20
Nomifensine	735.32	765.53	481.13	538.79	874.37	921.48
Valproate	457.79	502.32	252.95	301.40	620.30	789.30
Risperidone	1068.52	1083.19	822.57	825.30	1034.63	1081.01
Buspirone	1045.74	765.53	850.24	538.79	1065.22	921.48
Gabapentin	643.35	584.01	465.83	375.07	784.09	830.32
Dopamine	643.09	556.78	512.54	350.51	800.17	816.65
L-tyrosine	730.42	602.16	557.69	391.44	831.85	839.44
L-theanine	702.27	547.70	563.22	342.33	872.17	812.09
Mianserin	713.52	838.13	478.58	604.28	865.92	957.95
Folate	1500.13	1065.04	1358.86	808.93	1399.64	1071.90
Meprobamate	663.79	593.08	444.29	383.26	755.19	834.88
Amphetamine	504.29	529.55	285.37	325.96	700.68	802.97
Dehydroepiandrosterone	837.91	901.67	539.19	661.58	850.34	989.85
Pregnenolone	865.05	947.05	539.20	702.51	849.27	1012.64
Citalopram	873.22	883.52	571.52	645.21	890.36	980.74
Paroxetine	833.56	906.21	893.48	665.68	911.40	992.13
Duloxetine	805.31	838.13	536.28	604.28	906.24	957.95
Venlafaxine	674.87	751.91	405.77	526.51	809.54	914.65
Desvenlafaxine	738.02	738.30	504.79	514.24	817.09	907.81
Levomilnacipran	750.48	756.45	516.93	530.61	857.50	916.92
Clomipramine	754.45	847.21	479.31	612.47	861.87	962.50
Nortriptyline	597.85	643.00	360.07	428.28	803.75	859.95
Mirtazapine	691.85	702.00	566.66	481.49	853.81	889.58
Vilazodone	694.20	711.07	467.86	489.68	840.85	894.13
Tranylcypromine	483.20	574.93	279.51	366.89	734.59	825.76
Esketamine	682.82	692.92	440.65	473.31	847.57	885.02
Agomelatine	719.40	738.30	537.19	514.24	871.67	907.81
Moclobemide	754.38	729.22	548.22	506.05	928.78	903.25
Clorgiline	594.50	629.39	379.97	416.00	810.89	853.11
Tianeptine	713.06	720.15	437.54	497.86	833.21	898.69

Table 6.

Comparison of actual values and linear regression-predicted values for critical volume (CV) and molar refractivity (MR).

Drugs	Properties
	CV		MR
Structures Name	Actual	Predicted	Actual	Predicted
Bupropion	728.50	668.80	68.70	65.33
Amitriptyline	898.50	849.97	93.40	84.89
Effexor	859.50	801.65	83.18	79.68
Fluoxetine	861.50	849.97	81.04	84.89
Nefazodon	1308.50	1212.31	132.29	124.01
Paxil	888.50	970.75	90.51	97.93
Sertraline	852.50	837.89	85.99	83.59
Trazodone	984.50	1031.14	106.26	104.45
Nomifensine	719.50	777.50	77.56	77.07
Valproate	492.50	427.24	–	39.25
Risperidone	1144.50	1200.23	114.21	122.71
Buspirone	1117.50	777.50	108.44	77.07
Gabapentin	523.50	535.94	45.22	50.99
Dopamine	436.50	499.70	42.63	47.08
L-tyrosine	494.50	560.09	46.62	53.60
L-theanine	502.50	487.63	42.53	45.77
Mianserin	786.50	874.12	85.92	87.50
Folate	1186.50	1176.07	–	120.10
Meprobamate	638.50	548.02	53.73	52.29
Amphetamine	454.50	463.47	43.69	43.17
Dehydroepiandrosterone	893.50	958.67	84.02	96.63
Pregnenolone	1003.50	1019.06	93.11	103.15
Citalopram	977.50	934.51	94.41	94.02
Paroxetine	885.50	964.71	92.57	97.28
Duloxetine	864.50	874.12	89.70	87.50
Venlafaxine	846.50	759.38	82.01	75.11
Desvenlafaxine	701.50	741.26	68.50	73.16
Levomilnacipran	767.50	765.42	75.32	75.77
Clomipramine	960.50	886.20	97.93	88.81
Nortriptyline	612.50	614.44	61.17	59.47
Mirtazapine	588.50	692.95	59.84	67.94
Vilazodone	693.50	705.03	71.85	69.25
Tranylcypromine	455.50	523.86	47.78	49.69
Esketamine	650.50	680.87	70.88	66.64
Agomelatine	656.50	741.26	68.51	73.16
Moclobemide	733.50	729.19	74.24	71.85
Clorgiline	549.50	596.33	56.24	57.51
Tianeptine	734.50	717.11	69.75	70.55

Fig. 4.

Scatter plots showing the correlation between the most predictive topological indices and the experimental physicochemical properties.

Random forest regression

To capture the potential non-linear relationships between the topological indices and the physicochemical properties, we employed a Random Forest (RF) regression algorithm. Random Forest is an ensemble learning method that operates by constructing a multitude of de-correlated decision trees during training and outputting the mean prediction of the individual trees for regression tasks ²². This approach effectively mitigates overfitting by introducing randomness through bootstrap aggregating (bagging) of data samples for each tree and random subset selection of features for splitting nodes.

The RF model was trained using the eleven topological indices as the input feature vector p to predict each of the five target physicochemical properties. The ensemble prediction for a property y is given by:

2

where y is the predicted value, M is the total number of trees in the forest, and is the prediction of the i-th decision tree for the input feature vector p (containing the topological indices), and the summation is over all trees.

A key advantage of the Random Forest algorithm is its ability to quantify the relative importance of each feature in the predictive model. The importance is calculated as the total decrease in node impurity, measured by the Mean Squared Error (MSE), weighted by the probability of reaching that node, averaged over all trees.

The computed feature importance for each property are presented in Table 12. Analysis of this table reveals that the predictive utility of a topological index is highly property-specific, reflecting its ability to capture relevant aspects of molecular structure. Notably, the First Zagreb Index () and the Multiplicative Randic Index (MR(G)) emerged as consistently powerful predictors across multiple properties, particularly for Critical Volume (CV) and Molar Refractivity (MR). This suggests a strong link between these degree-based connectivity indices and properties influenced by molecular size and electron density. The Multiplicative Sombor Index () also demonstrated significant importance, especially for Boiling Point (BP) and Melting Point (MP), indicating its sensitivity to structural features governing intermolecular forces.

Table 12.

Feature importance scores of the topological indices for predicting each physicochemical property, as determined by the Random Forest algorithm.

Feature	BP	MP	CT	CV	MR
	0.103	0.118	0.072	0.098	0.053
	0.035	0.020	0.017	0.009	0.004
	0.084	0.079	0.050	0.039	0.016
	0.043	0.059	0.082	0.011	0.018
	0.036	0.034	0.035	0.021	0.017
	0.012	0.026	0.011	0.006	0.005
	0.169	0.232	0.107	0.175	0.146
	0.135	0.147	0.302	0.211	0.275
	0.149	0.114	0.135	0.260	0.250
	0.107	0.076	0.113	0.062	0.103
	0.125	0.095	0.075	0.109	0.113

The superior predictive performance of the First Zagreb index () and the Multiplicative Sombor index () can be interpreted in structural terms. The First Zagreb index, defined as the sum of the squares of vertex degrees, effectively quantifies the overall molecular branching and connectivity density. In the context of drug molecules, this correlates strongly with properties like molecular volume, surface area, and polarizability, which directly influence boiling points, critical volumes, and molar refractivity. Conversely, the Multiplicative Sombor index, based on the Euclidean norm of vertex degrees (), provides a more nuanced descriptor that is sensitive to the heterogeneity of the atomic environment within a molecule. Its product-based formulation amplifies its sensitivity to localized structural features and steric effects, making it particularly powerful for capturing complexities that govern solid-state properties like melting point and interactions related to critical temperature.

The predictive performance of the trained Random Forest model is comprehensively summarized in Table 14. The model achieved high accuracy, with coefficients of determination () ranging from 0.9154 (Melting Point) to 0.9791 (Critical Volume). The low error metrics, particularly for Molar Refractivity (MAE = 2.757, RMSE = 4.372), confirm the model’s robustness.

Table 14.

Performance metrics (MAE, MSE, RMSE, ) of the Random Forest regression model for each property.

Drugs	MAE	MSE	RMSE	r2
BP	22.441	1557.217	39.462	0.9536
MP	36.350	3300.074	57.446	0.9154
CT	20.216	1070.460	32.718	0.9300
CV	25.044	968.141	31.115	0.9791
MR	2.757	19.111	4.372	0.9561

A detailed comparison between the actual experimental values and the RF predicted values for all 38 compounds is cataloged in Table 13. The close agreement between these values demonstrates the model’s exceptional ability to generalize across a structurally diverse set of antidepressant drugs. Figures 5, 6, 7, 8, and 9 illustrate representative decision trees from the ensemble for each of the five target properties. These trees exemplify the simple, rule based logic employed at each node, showcasing how the model recursively partitions the feature space based on the topological indices to arrive at a prediction.

Table 13.

Comparison of actual values and random forest-predicted values for all five physicochemical properties.

Drugs	Properties
	BP		MP		CT		CV		MR
	Actual	Predicted	Actual	Predicted	Actual	Predicted	Actual	Predicted	Actual	Predicted
Bupropion	666.50	669.627550	394.64	412.839575	778.39	798.127525	728.50	691.287500	68.70	67.464000
Amitriptyline	751.01	751.602875	457.55	460.507125	856.98	857.196500	898.50	882.562500	93.40	90.687600
Effexor	766.61	756.954875	451.59	461.094350	812.60	830.249975	859.50	821.275000	83.18	81.508850
Fluoxetine	713.63	734.056675	410.29	436.292825	803.42	828.988325	861.50	865.907500	81.04	84.567275
Nefazodon	1092.30	1119.777650	812.28	869.038825	1072.62	1095.743125	1308.50	1242.495000	132.29	124.284350
Paxil	853.05	853.059325	600.10	669.315850	907.55	903.355775	888.50	906.720000	90.51	91.533875
Sertraline	788.23	768.149775	493.93	473.718375	872.67	858.505925	852.50	860.235000	85.99	85.806300
Trazodone	919.92	932.176625	716.19	730.724950	994.91	995.068800	984.50	995.592500	106.26	104.006275
Nomifensine	735.32	742.031650	481.13	478.367100	874.37	858.365450	719.50	751.855000	77.56	78.592300
Valproate	457.79	495.511025	252.95	295.570675	620.30	668.237550	492.50	484.242500	–	–
Risperidone	1068.52	1118.422250	822.57	888.288675	1034.63	1090.795375	1144.50	1171.090000	114.21	117.413950
Buspirone	1045.74	1010.537675	850.24	846.886075	1065.22	1056.589575	1117.50	1047.705000	108.44	106.625850
Gabapentin	643.35	621.692300	465.83	434.704250	784.09	775.325750	523.50	521.327500	45.22	46.965025
Dopamine	643.09	636.820800	512.54	475.292950	800.17	796.441725	436.50	460.497500	42.63	43.496250
L-tyrosine	730.42	675.451925	557.69	492.189375	831.85	808.475450	494.50	530.975000	46.62	48.860500
L-theanine	702.27	643.555875	563.22	487.172250	872.17	821.082800	502.50	483.542500	42.53	43.161825
Mianserin	713.52	737.346600	478.58	486.138225	865.92	870.562700	786.50	832.610000	85.92	88.874525
Folate	1500.13	1334.185650	1358.86	1149.858675	1399.64	1267.577525	1186.50	1180.590000	–	–
Meprobamate	663.79	654.061075	444.29	446.033750	755.19	768.603850	638.50	588.855000	53.73	50.869250
Amphetamine	504.29	558.308025	285.37	366.358800	700.68	746.524475	454.50	467.865000	43.69	43.645025
Dehydroepiandrosterone	837.91	846.614150	539.19	569.093925	850.34	869.242025	893.50	914.657500	84.02	87.301025
Pregnenolone	865.05	878.077075	539.20	640.780675	849.27	891.837425	1003.50	976.787500	93.11	93.994850
Citalopram	873.22	859.066125	571.52	572.511625	890.36	887.187850	977.50	954.222500	94.41	92.391625
Paroxetine	833.56	842.251000	893.48	779.958850	911.40	905.194150	885.50	898.622500	92.57	92.275750
Duloxetine	805.31	784.322575	536.28	515.318900	906.24	890.772725	864.50	873.275000	89.70	90.677900
Venlafaxine	674.87	696.230650	405.77	446.819550	809.54	830.148925	846.50	801.887500	82.01	78.531750
Desvenlafaxine	738.02	732.767325	504.79	507.961350	817.09	836.245050	701.50	709.465000	68.50	69.996625
Levomilnacipran	750.48	739.555150	516.93	498.045000	857.50	854.988700	767.50	773.130000	75.32	76.298900
Clomipramine	754.45	763.250875	479.31	491.701200	861.87	872.462125	960.50	917.445000	97.93	94.598350
Nortriptyline	597.85	611.737275	360.07	374.033525	803.75	802.148800	612.50	616.187500	61.17	61.904800
Mirtazapine	691.85	694.623950	566.66	529.455700	853.81	846.732825	588.50	629.627500	59.84	63.741125
Vilazodone	694.20	696.963250	467.86	481.428400	840.85	842.159200	693.50	679.492500	71.85	68.880225
Tranylcypromine	483.20	543.906975	279.51	345.239850	734.59	750.837400	455.50	487.535000	47.78	48.053500
Esketamine	682.82	688.528600	440.65	447.088275	847.57	845.176850	650.50	665.785000	70.88	69.754325
Agomelatine	719.40	720.417300	537.19	518.807550	871.67	862.763725	656.50	698.237500	68.51	71.269775
Moclobemide	754.38	734.984425	548.22	529.713800	928.78	894.372150	733.50	718.972500	74.24	71.889300
Clorgiline	594.50	617.446450	379.97	398.764525	810.89	806.356750	549.50	572.272500	56.24	58.145450
Tianeptine	713.06	716.202750	437.54	462.190850	833.21	840.597000	734.50	718.355000	69.75	70.153550

Fig. 5.

A representative decision tree from the random forest ensemble model for predicting boiling point.

Fig. 6.

A representative decision tree from the random forest ensemble model for predicting melting point.

Fig. 7.

A representative decision tree from the random forest ensemble model for predicting critical temperature.

Fig. 8.

A representative decision tree from the random forest ensemble model for predicting critical volume.

Fig. 9.

A representative decision tree from the random forest ensemble model for predicting molar refractivity.

The overall efficacy of the model is further visualized in Fig. 11 through violin plots, which compare the distribution of actual values against the distribution of predictions for each property. The strong overlap in the shapes and positions of these distributions indicates that the RF algorithm successfully captured the underlying data trends without introducing significant bias. Finally, Fig. 10 provides a consolidated comparison of the correlation coefficients (r) achieved by the RF model for each property, offering a clear visual synopsis of its predictive strength across all endpoints.

Fig. 11.

Violin plots comparing the distribution of actual values versus Random Forest predicted values for each physicochemical property.

Fig. 10.

Correlation coefficients comparison.

In conclusion, the Random Forest regression model effectively leveraged the information encoded in the topological indices to build a highly accurate, non-linear predictive framework. Its performance represents a substantial improvement over the linear regression baseline, unequivocally demonstrating the value of integrating graph-theoretic descriptors with advanced machine learning ensembles for QSPR modeling.

Extreme gradient boosting (XGBoost) regression

To further optimize predictive performance and push the boundaries of model accuracy, we implemented the Extreme Gradient Boosting (XGBoost) algorithm. XGBoost is a highly sophisticated and computationally efficient variant of gradient-boosted decision trees, renowned for its state-of-the-art results in machine learning competitions and scientific applications ²³. Unlike Random Forest, which builds trees in parallel, XGBoost employs a sequential, additive strategy where each new tree is constructed to correct the residual errors of the previous ensemble of trees. This iterative process of model refinement, combined with advanced regularization techniques to prevent overfitting, makes XGBoost exceptionally powerful for complex regression tasks.

The XGBoost model was trained using the same feature set of eleven topological indices to predict the five target properties. The core objective function that XGBoost optimizes is

where l is a differentiable convex loss function (mean squared error in our case) measuring the difference between the actual value and the prediction from the previous iteration, is the new tree added at step t, and is a regularization term that penalizes the complexity of the model to ensure generality.

The predictive prowess of the XGBoost model is unequivocally demonstrated by the error metrics summarized in Table 16. The algorithm achieved near perfect fits, with coefficients of determination () exceeding 0.99 for all five properties. Crucially, all error metrics Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) were significantly lower than those produced by both the linear regression and Random Forest models. For instance, for Boiling Point prediction, XGBoost achieved an RMSE of 12.40 and an of 0.995, compared to the Random Forest’s RMSE of 39.46 and of 0.954. This represents a dramatic improvement in predictive precision.

Table 16.

Performance metrics (MAE, MSE, RMSE, ) of the XGBoost regression model for each property.

Drugs	MAE	MSE	RMSE	r2
BP	8.59553	153.7925	12.40131	0.9954
MP	11.70513	278.1840	16.67885	0.9929
CT	8.22184	125.5067	11.20298	0.9918
CV	10.89980	207.6942	14.41160	0.9955
MR	0.78113	1.111332	1.05420	0.9976

A comprehensive comparison of the actual experimental values versus the values predicted by the optimized XGBoost model is presented in Table 15. The near one-to-one correspondence between these values for all 38 compounds underscores the model’s exceptional accuracy and its capacity to capture the underlying structure-property relationships with remarkable fidelity. The distribution of predictions is visually analyzed in Fig. 12 using violin plots. The near-perfect alignment between the shapes and densities of the actual and predicted value distributions for each property provides a powerful visual confirmation of the model’s lack of bias and its ability to accurately model the entire data distribution, not just its central tendency.

Table 15.

Comparison of actual values and XGBoost-predicted values for all five physicochemical properties.

Graphs	Properties
	Drugs	BP		MP		CT		CV		MR
Structures Name	Actual	Predicted	Actual	Predicted	Actual	Predicted	Actual	Predicted	Actual	Predicted
Bupropion	666.50	666.5317	394.64	394.6552	778.39	778.6478	728.50	727.9913	68.70	68.6869
Amitriptyline	751.01	751.0388	457.55	457.5875	856.98	857.0193	898.50	897.9124	93.40	93.3821
Effexor	766.61	766.1779	451.59	451.7574	812.60	812.6411	859.50	859.6270	83.18	83.1761
Fluoxetine	713.63	713.8526	410.29	410.6014	803.42	803.6005	861.50	861.4367	81.04	81.0400
Nefazodon	1092.30	1092.2930	812.28	812.4066	1072.62	1072.5872	1308.50	1308.5006	132.29	132.2820
Paxil	853.05	853.0968	600.10	600.0571	907.55	907.5279	888.50	888.9536	90.51	90.5134
Sertraline	788.23	788.2437	493.93	493.8382	872.67	872.6605	852.50	852.5325	85.99	85.9913
Trazodone	919.92	919.9562	716.19	716.2708	994.91	994.9431	984.50	984.6302	106.26	106.2504
Nomifensine	735.32	735.7570	481.13	481.0310	874.37	873.9747	719.50	720.0745	77.56	77.5652
Valproate	457.79	457.9774	252.95	253.1431	620.30	620.4323	492.50	492.2214	–	43.6822
Risperidone	1068.52	1068.6509	822.57	822.5342	1034.63	1034.7020	1144.50	1144.7823	114.21	114.2194
Buspirone	1045.74	1045.7694	850.24	850.2981	1065.22	1065.2697	1117.50	1117.6051	108.44	108.4407
Gabapentin	643.35	643.4077	465.83	465.8810	784.09	784.1264	523.50	523.4375	45.22	45.2176
Dopamine	643.09	643.0272	512.54	512.4622	800.17	800.1103	436.50	436.9966	42.63	42.6489
L-tyrosine	730.42	729.9401	557.69	557.2011	831.85	831.5355	494.50	495.1339	46.62	46.6413
L-theanine	702.27	701.9200	563.22	562.8556	872.17	871.9004	502.50	502.1482	42.53	42.5286
Mianserin	713.52	713.5710	478.58	478.6441	865.92	865.9796	786.50	787.1010	85.92	85.9206
Folate	1500.13	1499.7179	1358.86	1358.4829	1399.64	1399.4009	1186.50	1186.0114	–	107.2616
Meprobamate	663.79	663.8794	444.29	444.3555	755.19	755.2429	638.50	637.9984	53.73	53.7134
Amphetamine	504.29	504.3177	285.37	285.4157	700.68	700.7059	454.50	454.8996	43.69	43.6822
Dehydroepiandrosterone	837.91	838.1828	539.19	539.1700	850.34	850.3903	893.50	893.5867	84.02	84.0351
Pregnenolone	865.05	865.0406	539.20	539.5748	849.27	849.5446	1003.50	1003.5129	93.11	93.1070
Citalopram	873.22	872.8798	571.52	571.4445	890.36	890.2676	977.50	976.8000	94.41	94.4051
Paroxetine	833.56	833.4883	893.48	893.2295	911.40	911.2128	885.50	885.3615	92.57	92.5721
Duloxetine	805.31	804.9014	536.28	535.7065	906.24	905.8825	864.50	864.5922	89.70	89.7206
Venlafaxine	674.87	675.2484	405.77	406.2054	809.54	809.8925	846.50	846.6295	82.01	81.9891
Desvenlafaxine	738.02	738.0596	504.79	504.8771	817.09	817.1414	701.50	701.4644	68.50	68.4998
Levomilnacipran	750.48	750.0231	516.93	516.4423	857.50	857.5543	767.50	766.6555	75.32	75.3423
Clomipramine	754.45	754.8616	479.31	479.7987	861.87	862.1710	960.50	960.6238	97.93	97.9124
Nortriptyline	597.85	597.8602	360.07	359.9925	803.75	803.7602	612.50	612.5038	61.17	61.1695
Mirtazapine	691.85	691.8172	566.66	566.3538	853.81	853.7427	588.50	588.9933	59.84	59.8599
Vilazodone	694.20	694.2600	467.86	467.8459	840.85	840.8160	693.50	693.5289	71.85	71.8490
Tranylcypromine	483.20	483.5710	279.51	279.8251	734.59	734.8457	455.50	455.4069	47.78	47.7764
Esketamine	682.82	682.8347	440.65	440.6061	847.57	847.5918	650.50	650.4558	70.88	70.8773
Agomelatine	719.40	719.4717	537.19	537.2916	871.67	871.7058	656.50	657.1031	68.51	68.5067
Moclobemide	754.38	753.9449	548.22	547.9759	928.78	928.4456	733.50	732.7550	74.24	74.2360
Clorgiline	594.50	594.7835	379.97	380.4126	810.89	810.8322	549.50	549.6376	56.24	56.2416
Tianeptine	713.06	713.0833	437.54	437.8108	833.21	833.2880	734.50	734.6199	69.75	69.7550

Fig. 12.

Violin plots comparing the distribution of actual values versus XGBoost predicted values for each physicochemical property.

Comparative model analysis

A direct comparison of the error metrics from all three employed algorithms (Random Forest, and XGBoost) is graphically presented in Fig. 13 via a heatmap. This visualization allows for an immediate and intuitive assessment of relative model performance. The results clearly establish a performance hierarchy: XGBoost consistently outperformed Random Forest, which in turn significantly outperformed the linear regression baseline.

Fig. 13.

Heatmap comparing the error metrics (MAE, MSE, RMSE, ) of Random Forest, and XGBoost models across all properties.

This superiority can be attributed to XGBoost’s inherent advantages: its gradient boosting framework is specifically designed to minimize errors sequentially, its regularization controls complexity, and its handling of sparse data is highly efficient. These features make it ideally suited for learning the complex, non-linear mappings between the multidimensional topological descriptor space and the physicochemical properties.

Final comparisons between the actual values and the predictions from both ensemble methods are displayed in Figs. 14 and 15 for Random Forest and XGBoost, respectively. The superior convergence of the XGBoost-predicted data points (Fig. 15) onto the ideal line, compared to the greater spread seen in the Random Forest results (Fig. 14), offers a definitive visual proof of its enhanced predictive capability.

Fig. 14.

Scatter plots of actual vs. random forest-predicted values for all five properties.

Fig. 15.

Scatter plots of actual vs. XGBoost-predicted values for all five properties.

In summary, the XGBoost algorithm demonstrated itself as the most powerful predictive tool in this study. Its implementation resulted in a highly accurate, robust, and generalizable QSPR model that effectively translates graph-theoretic molecular descriptors into precise estimations of critical physicochemical properties, setting a new benchmark for computational efficiency and predictive accuracy in this domain.

Data splitting and validation strategy

To reduce overfitting and obtain an unbiased estimate of predictive performance, we used a deterministic 80/20 train test split on the full set of 38 antidepressant drugs. The training set contains the drugs from Bupropion to Nortriptyline, and the blind test set contains the drugs from Mirtazapine to Tianeptine. The blind set was kept completely unseen during descriptor processing, model fitting, and any performance based selection. Each drug was described by the 11 degree-based topological indices listed in Table 3. The predicted properties were boiling point (BP), melting point (MP), critical temperature (CT), critical volume (CV), and molar refractivity (MR) (see Table 4).

Model development was performed exclusively on the training set using 5-fold cross-validation (i.e., ). For both the Random Forest and XGBoost algorithms, we utilized the default hyperparameters provided by their standard implementations (in Python’s scikit-learn and xgboost libraries, respectively), and no additional hyperparameter tuning was conducted. This approach was chosen to ensure a straightforward comparative analysis and to mitigate the risk of overfitting given the dataset size. We report fold-averaged , RMSE, and MAE in Table 17. After model selection, the models were refit on the full training set and evaluated once on the held-out blind test (Table 18). Two learners were considered Random Forest and XGBoost. No tuning was performed on the test set, and no information from the blind set was used during cross-validation or model selection. The outcomes from training cross-validation and blind testing align for MR and CV, indicating that the degree-based indices capture structural features relevant to molar refractivity and critical volume. These results provide a consistent account across cross-validated training measures and blind-test measures.

Table 17.

Five-fold cross-validation (training-only) mean performance with topological indices.

Property	Model		RMSE	MAE
BP	Random Forest	0.750	84.700	62.068
BP	XGBoost	0.500	111.324	74.347
CT	Random Forest	-0.066	73.379	59.693
CT	XGBoost	-0.017	86.582	67.062
CV	Random Forest	0.844	76.953	66.489
CV	XGBoost	0.807	83.932	72.394
MP	Random Forest	-0.052	131.278	100.811
MP	XGBoost	-0.492	160.337	114.932
MR	Random Forest	0.844	7.812	6.510
MR	XGBoost	0.839	7.658	6.337

Table 18.

Blind test (held-out Mirtazapine to Tianeptine) performance with topological indices.

Property	Model		RMSE	MAE
BP	Random Forest	0.226	71.715	42.348
BP	XGBoost	0.003	81.395	61.969
CT	Random Forest	-0.057	52.838	45.238
CT	XGBoost	-1.157	75.486	67.979
CV	Random Forest	0.064	87.482	75.958
CV	XGBoost	0.495	64.238	48.750
MP	Random Forest	-0.073	93.354	72.917
MP	XGBoost	-1.156	132.342	112.829
MR	Random Forest	0.645	5.166	3.633
MR	XGBoost	0.636	5.233	4.158

Concluding remarks

This study successfully demonstrates the potent synergy between graph-theoretic molecular descriptors and modern machine learning algorithms for advanced Quantitative Structure-Property Relationship (QSPR) modeling. We computed a comprehensive set of eleven topological indices, encompassing both classical and novel multiplicative variants, for 38 diverse antidepressant compounds. These indices served as robust numerical descriptors encoding key aspects of molecular structure.

Our investigation reveals that while simple linear regression models established statistically significant baseline relationships with the First Zagreb index () emerging as a particularly strong linear predictor their performance was substantially surpassed by non-linear ensemble methods. The Random Forest algorithm provided a significant boost in predictive accuracy by effectively capturing complex, non-linear interactions within the descriptor space. Ultimately, the Extreme Gradient Boosting (XGBoost) algorithm delivered state-of-the-art performance, achieving exceptional predictive accuracy for all five physicochemical properties boiling point, melting point, critical temperature, critical volume, and molar refractivity as evidenced by near-perfect coefficients of determination () and minimal error metrics (e.g., for boiling point).

While this study demonstrates a powerful and highly accurate predictive framework, it is important to acknowledge that the model was trained on a dataset of 38 antidepressant drugs. Although this represents a meaningful expansion from earlier studies and encompasses significant structural diversity, the dataset size remains a limitation from a machine learning perspective. The exceptional performance metrics (e.g., ) indicate a strong learned relationship, but the generalizability of the model would be further strengthened by applying this methodology to larger and more diverse chemical datasets in future work. Such expansion would serve to rigorously validate the robustness of the identified structure-property relationships and enhance the model’s predictive power across a broader chemical space.

This work underscores that multiplicative topological indices are not merely mathematical abstractions but practical and powerful tools for computational pharmaceutical science. The proposed interdisciplinary framework, integrating concepts from chemical graph theory, statistics, and machine learning, offers a highly accurate and efficient strategy for predicting critical drug properties. This approach can significantly accelerate the preliminary stages of drug discovery and design by reducing the reliance on costly and time-consuming experimental measurements. Future work will focus on extending this methodology to a broader range of compound classes and biological endpoints, further validating its utility as a cornerstone of modern computational chemistry.

Author contributions

Conceptualization: S. N, S. A, M. E. M. A, M. E. D, A. A, K. A. T Methodology: S. N, S. A, A. A Formal Analysis: G. Z, S. A, M. E. M. A, M. E. D, K. A. T Validation: G.Z, S. N, S. A, M. E. M. A, M. E. D, A. A, K. A. T Writing original draft: S. N, S. A, A. A Writing review and editing: G. Z, M. E. M. A, M. E. D, K. A. T

Data availability

All data generated or analysed during this study are included in this published article

Declarations

Competing interests

The authors declare no competing interests.