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QSPR analysis of some novel neighborhood degree based topological descriptors
Sourav Mondal, Nilanjan De, Anita Pal
TL;DR
The paper addresses the need for molecular descriptors that capture neighborhood degree information for correlating chemical structure with physical and chemical properties. It proposes six such indices, supplies an algorithm for their computation, and evaluates them through QSPR analyses, correlation studies, and degeneracy testing. The indices are reported as useful QSPR descriptors, with performance varying by property and index; ND5 is identified as independent among the studied indices.
Problem
Existing topological indices can suffer degeneracy, where different isomers share the same value, motivating evaluation of new descriptors based on neighborhood degree sums.
Method
The paper proposes six neighborhood degree-sum indices, designs an algorithm to compute them, and tests them using regression, correlation, and degeneracy analyses.
Results
The indices are useful molecular descriptors for QSPR research, with predictive power sometimes superior and sometimes inferior to established indices; ND5 is independent among the studied indices.
Takeaways & Limitations
Neighborhood degree-sum indices extend established degree-based descriptors and can be applied to QSPR modeling and isomer discrimination within the studied datasets.
Abstract
from arXiv · showhide
Topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this work, some new indices based on neighbourhood degree sum of nodes are proposed. To make the computation of the novel indices convenient, an algorithm is designed. QSPR analysis of these newly introduced indices are studied here which reveals their predicting power. Some mathematical properties of these indices are also discussed here.
1. Introduction
The paper introduces six neighborhood degree-sum indices as extensions of degree-based topological descriptors and frames them for chemical-structure modeling. It motivates their use in QSPR analysis and mathematical comparison with established indices.
- Topological indices encode structural information in molecular graphs and support modeling physical and chemical properties through QSPR/QSAR analysis.
- The indices are defined for molecular graphs, where vertices represent atoms, edges represent chemical bonds, and dG(u), dG(v) denote vertex degrees.
- The paper introduces six neighborhood degree-sum indices named ND1 through ND6 as a continuation of earlier neighborhood-degree indices.
- The study evaluates chemical applicability through computation, regression-based QSPR analysis, degeneracy testing, comparison with established indices, and mathematical relations.
2. Computational aspects
The computational procedure represents graph connectivity, vertex degrees, and neighborhood degree sums in matrices. The novel indices are computed as functions of endpoint degrees and their neighborhood degree sums.
- The algorithm is designed to make computation of the novel indices convenient.
- The procedure takes a graph as input and calculates vertex degrees and neighborhood degree sums using connectivity and degree-related arrays.
- The conn[E][2] matrix stores vertex connections, while deg[V][2] and δ[V][2] store vertex degrees and neighborhood degree sums.
- Each novel index can be expressed as a function of δG(u), δG(v), dG(u), and dG(v).
3. Newly introduced indices in QSPR analysis
The QSPR analysis applies the newly designed indices to properties of octane isomers and 67 alkanes. It examines their ability to model physicochemical and physical properties.
- The analysis models five physicochemical properties of octane isomers and six physical properties of 67 alkanes ranging from n-butanes to nonanes.
- The octane-isomer properties include acentric factor, entropy, enthalpy of vaporization, standard enthalpy of vaporization, and heat capacity at constant pressure.
- The alkane properties include boiling points, molar volumes, molar refractions, heats of vaporization, surface tensions, and melting points.
Regression model for octane isomers:
The paper specifies linear QSPR regression models relating neighbourhood degree-based indices to measured properties of octane isomers, with statistical parameters reported for ND1 through ND6. The accompanying figures depict correlations between each index and selected octane-isomer properties.
- Model specification: The regression model represents each physical property as P = m(TI) + c, where TI is a topological index.The supplied model description identifies P as the physical property and TI as the topological index.
- Statistical evaluation: The reported model statistics include intercept, slope, correlation coefficient, standard error, F-test, and significance F.Correlation coefficient measures linear association, while significance F below 0.05 is identified as indicating statistical significance.
- Model specification: Statistical tables report linear-model parameters for ND1, ND2, ND3, ND4, ND5, and ND6.The tables are titled as statistical parameters for the linear QSPR model of each respective index.
- Octane-isomer correlations: Figures 1–6 show correlations of ND1–ND6, respectively, with entropy, acentric factor, and DHVAP for octane isomers.Each figure caption assigns one neighbourhood degree-based index to the same three octane-isomer properties.
Regression model for 67 alkanes:
The paper evaluates neighbourhood degree-based indices as linear QSPR predictors for alkane properties and octane-isomer properties. Predictive quality varies substantially across indices and target properties.
- The regression models use intercept, slope, correlation coefficient, standard error, F-test, and significance F to assess linear QSPR relationships.
- ND1 correlates well with octane-isomer entropy, acentric factor, and DHVAP, especially acentric factor with r = −0.9904.
- ND2 shows strong alkane correlations except melting point, ranging from 0.809 to 0.9638 and reaching 0.9638 for hv.
- ND3 is inadequate for alkane structure-property correlations, with coefficients from 0.2036 to 0.7318, but correlates with octane-isomer entropy and acentric factor.
- ND4 correlates well with several octane-isomer properties but is generally inadequate for alkanes except cp and hv, with coefficients −0.8634 and 0.8679.
- ND5 predicts several alkane properties with coefficients from 0.8881 to 0.9779, whereas ND6 has poor alkane correlations ranging from 0.2192 to 0.7823.
4. Correlation with some well-known indices
The new indices are compared with established topological indices and with one another for octane isomers. Most show high correlations, while ND5 is comparatively independent.
- The new indices have high correlations with established indices except ND5.
- The highest reported correlation is r = 0.9977 between ND1 and M2.
- ND5 has significantly low correlations with the other new indices and is therefore characterized as independent among the five indices.
- Figure 7 represents indices as vertices, joining two vertices when |r|≥0.95.
5. Degeneracy
The paper treats degeneracy as a limitation of topological indices and evaluates the new indices using sensitivity-based isomer discrimination. The new indices show good response for octane and decane isomers.
- Degeneracy occurs when two or more isomers possess the same topological-index value, limiting structural discrimination.
- Sensitivity measures the proportion of isomers distinguished by an index, and higher sensitivity indicates greater isomer-discrimination power.
- The newly introduced indices exhibit good sensitivity among the investigated degree-based indices for octane and decane isomers.
6. Mathematical properties
The paper derives mathematical bounds for the new neighbourhood degree-based indices using standard inequalities. It identifies equality conditions and relates several new indices to established descriptors.
- The bounds are developed for simple connected graphs using standard inequalities and auxiliary sequence lemmas.
- Several propositions provide bounds involving neighbourhood Zagreb, hyper Zagreb, forgotten, geometric-arithmetic, and second Zagreb indices.
- For one bound, equality holds only when G is P2.
- Equality in multiple bounds holds when G is regular or a complete bipartite graph.
- A derived inequality is 2m^2 ≤ GA5(G)ND5(G).
- The paper establishes ND1(G) ≥ RR(G), ND2(G) ≥ SCI(G), ND3(G) ≥ ReZG3(G), and ND4(G) ≥ R(G).
7. Conclusion
The study proposes neighborhood-degree-based indices and evaluates their QSPR usefulness, degeneracy, correlations, and mathematical bounds. ND5 is identified as independent among the novel indices, while predictive performance varies against established descriptors.
- The indices were tested as molecular descriptors using octane isomers and 67 alkanes.
- Their predictive power was sometimes superior and sometimes inferior to established degree-based indices.
- The degeneracy test reported in Table 17 supported the new indices over the older indices.
- Correlation analyses identified ND5 as independent among all the novel indices.
- The study also derived bounds for the new indices and proposed evaluating them on graph operations, composite graphs, and networks.