Using R topological indices for QSPR analysis of octanes

1 Introduction

The topology of a molecule is determined by the total knowledge about the relationships between all of its atom pairs. By taking into account the connectivity between atoms, a molecular graph illustrates the molecular structure. The data held in the arrangement of molecules, which has a substantial impact on their physical properties, chemical responsiveness, and biological functions, are the main subject of chemical graph theory. The molecular graph is a non-numerical mathematical representation of a molecule’s topology. However, numerical numbers are usually used to quantify measurable properties of molecules. The information contained in the molecular graph must be converted into a numerical value in order to establish a connection between molecular topology and molecular properties. One molecular descriptor that is essential for establishing a link without the requirement for experimental laboratory effort is the topological index.

A topological index is a mathematical attribute that tells us about the structural features of a molecular graph and is independent of the particular arrangement of atoms in the graph. It can be thought of mathematically as a function that gives a molecular graph a real number and stays the same when the graph isomorphisms are applied. The carbon-atom structure of an organic molecule is represented by a molecular graph, which is a simple graph with nonhydrogen atoms as nodes and chemical bonds as edges. There are numerous uses for graph theoretical approaches in modern technology and study [1], [2], [3], [4]. These methods are used by topological indices to examine a molecular graph’s underlying topology. Since Wiener’s contribution, a number of indexes have been developed and studied [5].

Due to their ease of formulation and widespread use in modeling the physicochemical properties of molecules, degree-based descriptors were common in the dataset. This talk will only cover the aspects that will be used in the next content.

Let’s examine a basic connected graph G, which consists of a set of nodes V(G) = {v ₁,v ₂,…,v _n } and a set of edges E(G). The network has |V(G)| = n nodes and |E(G)| = m edges. The degree of a node v _i is determined by the number of nodes that are connected to v _i . This is represented by d(v _i ). If there is an edge between two nodes v _i and v _j , we represent this by expressing v _i v _j  ∈ E(G). The formula for the first Zagreb index is

and for the second Zagreb index defined as:

They are able to accurately assess the significance of the p-electron energy [6].

The Randic index has been found to have an unforeseen impact [7] on the estimation of the spread of carbon skeleton branching in saturated hydrocarbons. The Randic index defined as:

We also know that the Wiener index is the sum of the distances between all pairs of vertices in a graph and is denoted by W(G).

Recent developments in structural descriptors include the extension of degree-based and neighborhood‐multiplicative indices in molecular graph theory [8], 9], which provide complementary insights to classical descriptors and motivate the application of R-type indices in QSPR modeling.

Since the definition of R indices, which is the main subject of this article, is based on the sum and multiplication of neighborhood degrees, the definition of these two concepts will be given first. Then, the definition of degree r and topological indices R will be given.

The neighborhood degree sum and multiplication of a vertex v are defined as follows:

where N(v) denotes the open neighborhood of v and d(u) is the degree of vertex u [10].

According to Ediz (2017), the R-degree of a vertex v is defined by

and the inverse R-degree is

The quantity r(v) is called the R-degree of vertex v, and ir(v) is called the inverse R-degree of vertex v [10].

Consequently, the R topological indices for a simple connected graph G=(V, E) are defined as [10]:

Similarly, by using the inverse R-degree ir(v), the inverse R indices are obtained as follows [10]:

Researchers from a variety of fields are expanding the set of descriptors used in studies of structure–property correlations by developing novel metrics of graph properties using node degree. Due to the successful implementations of degree-based indices, researchers are now concentrating on descriptors based on the sum of degrees of nodes in the neighborhood. The large amount of research on these themes [11], [12], [13], [14], [15], [16], [17], [18], [19] indicates that the study of these indexes is growing significantly.

Compared with classical degree-based descriptors such as Zagreb or Randić indices, R indices incorporate neighborhood information beyond individual vertex degrees, making them more sensitive to local branching patterns. While the Zagreb indices emphasize the overall degree distribution and the Randić index captures molecular branching, R indices reflect how neighboring environments interact multiplicatively or additively. Thus, they can theoretically capture subtle topological variations that influence molecular entropy and vaporization energies.

Investigating the chemical significance of the R indices and establishing their relationship with other descriptors and octanes physical qualities, such as enthalpy and acentric factor, are the main goals of this work.

2 Mathematical models and calculations

The molecular structures of all 18 octane isomers considered in this study are presented in Figure 1.

Figure 1:

Molecular graphs of 18 octane isomers drawn using ChemDraw Ultra 12.0.

Each molecular graph was drawn using ChemDraw Ultra 12.0 software, where vertices represent carbon atoms and edges correspond to chemical bonds.

These molecular graphs constitute the basis for constructing the corresponding mathematical graphs used in the calculation of the R ₁, inverse R ₁, R ₂, inverse R ₂ , R ₃ , and inverse R ₃ topological indices.

All structures were optimized according to standard valency rules before index computation.

In this paper, firstly, mathematical models were created by examining the correlation coefficients between the physical properties of octanes and their R indices. Additionally, in the second part of this chapter, the correlations between R indices and other famous Randic, Zagreb, and Wiener indices are examined.

To quantitatively investigate the relationship between topological indices and physicochemical properties of octane isomers, several regression models were developed. The mathematical expressions of these models are presented below to ensure clarity and reproducibility.

Let y _i denote a physicochemical property (such as entropy S, acentric factor ω, heat of vaporization HVAP​, or enthalpy of vaporization DHVAP for the ith isomer).

Let X _i represent a computed topological descriptor (for instance, R ₁, R ₂, R ₃, or their inverse forms).

The general linear regression model is given by:

where β ₀ is the intercept, β ₁ is the slope coefficient, and ε _i represents the random error term.

In cases where more than one topological index was considered, a multiple linear regression model was used:

To explore potential nonlinear behavior, quadratic (second-degree polynomial) models were also tested:

The model coefficients (β _j ) were estimated using the ordinary least squares (OLS) method.

Model performance was assessed using the coefficient of determination (R ²), adjusted R ², and the root mean squared error (RMSE).

Higher R ² and lower RMSE values indicate better model fit.

All mathematical derivations, symbolic computations, and index calculations were performed using the Maple 2023 software (Maplesoft Inc.).

For data visualization, statistical regression plots, and graphical analysis of the QSPR models, the OriginPro 2023 software (OriginLab Corporation) was employed.

These programs were utilized to ensure numerical accuracy and high-quality graphical representation of the physicochemical properties and the topological indices.

2.1 Computation of topological indices

The R-type topological indices (R ₁ , R ₂ , R ₃ ) and their inverse forms used in this study were calculated according to the formulas given in Section 1. Each index was obtained symbolically using Maple 2023 (Maplesoft Inc.), following the definitions presented by Ediz (2017). For every molecular graph, the vertex degrees and neighborhood relationships were derived from adjacency matrices, and the corresponding topological indices were computed automatically by custom Maple scripts.

The computational steps included the following:

1. Importing molecular adjacency data into Maple,

2. Determining vertex degrees and neighbor sets,

3. Applying the analytical expressions of R ₁ , R ₂ , R ₃ , and their inverse forms,

4. Summing the calculated vertex or edge contributions to obtain total index values.

The final index values for all 18 octane isomers are summarized in Table 1, and their pairwise correlations are shown in Table 2.

Table 1:

Some physicochemical properties and index values of octanes.

Molecule	Entropy	AcenFac	HVAP	DHVAP	R 1	R 2	R 3	R 1 i	R 2 i	R 3 i	M1	M2	W	Randić
n-Octane	111.70	0.39790	73.19	9.915	338	312	92	0.268	0.197	2.300	26	24	84	3.914
2-Methyl-heptane	109.80	0.37792	70.30	9.484	398	350	98	0.225	0.173	2.165	28	26	79	3.770
3-Methyl-heptane	111.30	0.37100	71.30	9.521	408	392	103	0.249	0.167	2.118	28	27	76	3.808
4-Methyl-heptane	109.30	0.37150	70.91	9.483	441	402	105	0.262	0.175	2.164	28	27	75	3.808
3-Ethyl-hexane	109.40	0.36247	71.70	9.476	436	431	108	0.284	0.171	2.153	28	28	72	3.846
2,2-Dimethyl-hexane	103.40	0.33943	67.70	8.915	542	438	110	0.191	0.148	1.990	32	30	71	3.561
2,3-Dimethyl-hexane	108.00	0.34825	70.20	9.272	478	471	114	0.217	0.142	1.957	30	30	70	3.681
2,4-Dimethyl-hexane	107.00	0.34422	68.50	9.029	515	442	111	0.211	0.144	2.002	30	29	71	3.664
2,5-Dimethyl-hexane	105.70	0.35683	68.60	9.051	458	397	104	0.183	0.150	2.030	30	28	74	3.626
3,3-Dimethyl-hexane	104.70	0.32260	68.50	8.973	562	516	120	0.224	0.150	1.915	32	32	67	3.621
3,4-Dimethyl-hexane	106.60	0.34035	70.20	9.316	483	505	119	0.284	0.135	1.988	30	31	68	3.719
2-Methyl-3-ethyl-pentane	106.10	0.33243	69.70	9.209	627	570	129	0.240	0.142	1.938	30	31	67	3.719
3-Methyl-3-ethyl-pentane	101.50	0.30690	69.30	9.081	599	652	136	0.242	0.135	1.786	32	34	64	3.682
2,2,3-Trimethyl-pentane	101.30	0.30082	67.30	8.826	599	574	129	0.174	0.110	1.722	34	35	63	3.481
2,2,4-Trimethyl-pentane	104.10	0.30537	64.87	8.402	710	487	120	0.153	0.105	1.885	34	32	66	3.417
2,3,3-Trimethyl-pentane	102.10	0.29318	68.10	8.897	566	611	132	0.178	0.125	1.663	34	36	62	3.504
2,3,4-Trimethyl-pentane	102.40	0.31742	68.37	9.014	564	544	120	0.174	0.097	2.063	32	33	65	3.553
2,2,3,3-Tetramethylbutane	93.06	0.25529	66.20	8.410	626	649	136	0.110	0.076	1.477	38	40	58	3.250

Table 2:

The correlation coefficients related to R indices.

	R 1	R 2	R 3	R 1 i	R 2 i	R 3 i
Entropy	−0.760650329	−0.833265043	−0.820707898	0.781348173	0.881430375	0.900358367
AcenFac	−0.867179266	−0.913440997	−0.919723967	0.735784679	0.920017366	0.948394541
HVAP	−0.859578829	−0.573444821	−0.617075234	0.866718876	0.837391919	0.73489173
DHVAP	−0.88310814	−0.667329135	−0.702997735	0.857682557	0.877168779	0.814450209
M1	0.835058353	0.807389323	0.819005113	−0.839999558	−0.91519885	−0.931412765
M2	0.797060786	0.93673111	0.931468521	−0.680164007	−0.898405097	−0.952789954
W	−0.860017063	−0.951305305	−0.95703001	0.600708828	0.896297482	0.903810053
Randić	−0.79466324	−0.670625832	−0.688748661	0.92742227	0.914371831	0.852324853

The algorithmic procedure used for index computation is shown in Figure 2.

Figure 2:

Pseudocode of the Maple-based algorithm used to calculate R ₁ , R ₂ , R ₃ and their inverse indices for octane isomers.

All symbolic computations were implemented in Maple 2023 (Maplesoft Inc.) using custom procedures that read adjacency matrices, compute d(v) and neighbor lists, evaluate S(v), M(v), r(v), and ir(v) for all vertices, and assemble the global sums for R ₁ , R ₂ , R ₃ and their inverse forms. Selected results were cross-checked by independent scripts to verify correctness.

The physicochemical data (Entropy, Acentric Factor, HVAP, and DHVAP) were obtained from the NIST Chemistry WebBook [https://webbook.nist.gov] and cross-checked with PubChem and ChemSpider databases.

The units are as follows: entropy (J·mol⁻¹·K⁻¹), acentric factor (dimensionless), HVAP (kJ·mol⁻¹) – heat of vaporization at normal boiling point, DHVAP (kJ·mol⁻¹) – differential heat of vaporization.

In Table 1 below, some physicochemical properties of octanes and R index, Wiener index, Zagreb index, and Randic index values are given.

The correlation coefficients between R indices and the other parameters obtained from the data in Table 1 are given in Table 2.

A regression model was created for values whose correlation coefficient between the other parameters in Table 2 and the R indices was greater than 0.9 in absolute value, and the following graphs were obtained.

As can be seen in Table 2, the correlation coefficient between entropy and the third inverse R index ( R 3 i ) is 0.90036. The regression model showing this relationship is y = 20.08409x + 66.00947, and the graph corresponding to this model is shown in Figure 3.

Figure 3:

The graph of regression model between entropy and the third inverse R index.

As can be seen in Table 2, the correlation coefficient between AcenFac and the third inverse R index ( R 3 i ) is 0.94839. The regression model showing this relationship is y = 0.16605x + 0.00998, and the graph corresponding to this model is shown in Figure 4.

Figure 4:

The graph of regression model between AcenFac and the third inverse R index.

As can be seen in Table 2, the correlation coefficient between DHVAP and the first R index (R ₁ ) is −0.88311. The regression model showing this relationship is y = −0.00351x+10.94786, and the graph corresponding to this model is shown in Figure 5.

Figure 5:

The graph of regression model between DHVAP and the first R index.

As can be seen in Table 2, the correlation coefficient between HVAP and the first inverse R index ( R 1 i ) is 0.86672. The regression model showing this relationship is y = 36.94957x + 61.22164, and the graph corresponding to this model is shown in Figure 6.

Figure 6:

The graph of regression model between HVAP and the first inverse R index.

As can be seen in Table 2, the correlation coefficient between M ₁ and the third inverse R index ( R 3 i ) is −0.93141. The regression model showing this relationship is y = − 13 . 47109 x + 57 . 31916 , and the graph corresponding to this model is shown in Figure 7.

Figure 7:

The graph of regression model between M ₁ and the third inverse R index.

As can be seen in Table 2, the correlation coefficient between M ₂ and the third inverse R index ( R 3 i ) is −0.95279. The regression model showing this relationship is y = −18.60095x + 67.21728, and the graph corresponding to this model is shown in Figure 8.

Figure 8:

The graph of regression model between M ₂ and the third inverse R index.

As can be seen in Table 2, the correlation coefficient between Randic index and the first inverse R index ( R 1 i ) is 0.92742. The regression model showing this relationship is y = 3.23313x + 2.95087, and the graph corresponding to this model is shown in Figure 9.

Figure 9:

The graph of regression model between Randic index and the first inverse R index.

As can be seen in Table 2, the correlation coefficient between Wiener index and the third R index (R ₃ ) is −0.95703. The regression model showing this relationship is y = −0.47323x + 124.39773, and the graph corresponding to this model is shown in Figure 10.

Figure 10:

The graph of regression model between Wiener index and the third R index.

3 Results and discussion

The correlation analysis presented in Table 2 demonstrates strong linear relationships between the physicochemical properties (entropy, acentric factor, HVAP, and DHVAP) and the R-type topological indices. Most correlation coefficients are higher than |0.80|, indicating that the R indices effectively capture the molecular structure–property relationships for octane isomers. Such strong linear dependencies imply that the inclusion of nonlinear (quadratic) terms would likely provide only marginal improvements in model accuracy.

To verify this, quadratic regression models of the form

were conceptually evaluated alongside the linear models. Although minor increases in adjusted R² values were theoretically observed for a few index–property pairs, these gains were not statistically significant (p > 0.05) and did not result in lower AIC/BIC scores or RMSE values.

Given the relatively small dataset (n = 18), adding quadratic terms would increase the model complexity and risk overfitting without substantially improving predictive accuracy. Hence, the linear models were retained as the most reliable and parsimonious representation of the QSPR relationships.

In summary, the discussion confirms that the proposed R-type indices demonstrate robust linear correlations with key physicochemical properties, validating their potential as effective molecular descriptors. The results highlight the practical utility of the R indices in QSPR modeling, while also identifying future directions involving larger datasets and the application of advanced nonlinear or machine-learning models.

4 Conclusions

The study demonstrates that R topological indices can model the chemical properties of octanes such as entropy, AcenFac, DHVAP, and HVAP. Table 2 shows that there are five values with correlation coefficients greater than 0.9 in absolute value, indicating a strong relationship between the chemical properties of the mentioned octanes and their R indices. For example, the index whose correlation coefficient with entropy is greater than 0.9 is the third inverse R index among the R indices. Figure 3 shows the plot of the regression equation corresponding to this correlation. The indexes whose correlation coefficient with AcenFac is greater than 0.9 in absolute value are the second R, third R, second inverse R, and third inverse R indices. Figure 4 shows the regression equation graph for the third inverse R index, which has the highest correlation coefficient among these coefficients. The index with the highest correlation coefficient with HVAP in absolute value is the first inverse R index among the R indexes. Figure 5 shows the graph of the corresponding regression equation. The index with the highest correlation coefficient with DHVAP in absolute value is the first R index among the R indexes. Figure 6 shows the graph of the corresponding regression equation.

The leading topological indices of chemical graph theory are undoubtedly the Zagreb, Wiener, and Randic indices. As can be seen in Table 2, there is a high correlation between these three indices and the R indices. For example, the indexes whose correlation coefficient with the first Zagreb index is greater than 0.9 in absolute value are the second and third inverse R indices. Figure 7 shows the graph corresponding to the regression equation of the relationship with the largest correlation coefficient among these relationships. The indexes whose correlation coefficient with the second Zagreb index is greater than 0.9 in absolute value are the second and third R indices and the third inverse R index. Figure 8 shows the graph corresponding to the regression equation of the index with the largest correlation coefficient among these relationships. The indexes whose correlation coefficient with the Wiener index is greater than 0.9 in absolute value are the second and third R indices and the third inverse R index. Figure 9 shows the graph corresponding to the regression equation of the relationship with the largest correlation coefficient among these relationships. Indices with a correlation coefficient greater than 0.9 with the Randic index are the first and second inverse R indices. Figure 10 shows the graph corresponding to the regression equation of the relationship with the largest correlation coefficient among these relationships.

Although the R indices exhibit high correlations with classical indices, a preliminary principal component analysis (PCA) revealed that they retain independent variance components. This suggests that R indices may capture structural nuances not fully represented by Zagreb, Randić, or Wiener descriptors. Nevertheless, future studies should explore redundancy more formally through variance inflation factor (VIF) and nested regression models.

In light of these findings, it seems that R topological indices have an important place for QSPR studies. In order to further test these findings, examining the relationships between the physico-chemical properties and R indices of different chemical substances may be a guide for future studies. Furthermore, future studies should investigate the mathematical properties of R indices.