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arxiv: 2606.05989 · v2 · pith:5PY5LOQEnew · submitted 2026-06-04 · ⚛️ physics.chem-ph

A Comparative Study of Exponential Sum-Connectivity and Product-Connectivity Gourava Indices for Benzenoid Hydrocarbons

H. M. Nagesh , B. Azghar Pasha , U. Vijaya Chandra Kumar , Narahari N This is my paper

Pith reviewed 2026-06-27 23:25 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords exponential Gourava indicesbenzenoid hydrocarbonssum-connectivityproduct-connectivityπ-electronic energiesQSPRregression analysismolecular graphs
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The pith

Exponential product-connectivity Gourava index slightly outperforms the sum-connectivity variant in predicting π-electronic energies of benzenoid hydrocarbons with correlations above 0.999

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the exponential sum-connectivity Gourava index and the exponential product-connectivity Gourava index for benzenoid hydrocarbons. These descriptors show strong mutual correlation and are evaluated through regression as predictors of π-electronic energies. Both achieve correlation coefficients exceeding 0.999, with the product-connectivity version providing coefficients that align more closely with optimal least-squares results. The findings position the indices as a framework for modeling structural characteristics and electronic properties in these molecular systems.

Core claim

The exponential sum-connectivity Gourava index (e^{SGO(G)}) and the exponential product-connectivity Gourava index (e^{PGO(G)}) are computed for benzenoid hydrocarbons. Regression analysis shows both are exceptionally reliable predictors of π-electronic energies, with correlation coefficients exceeding 0.999. The product-connectivity variant offers a slightly superior fit because its coefficients align more precisely with optimal least-squares results.

What carries the argument

The exponential sum-connectivity Gourava index (e^{SGO(G)}) and exponential product-connectivity Gourava index (e^{PGO(G)}), computed from molecular graphs to capture structural characteristics and serve as predictors of π-electronic energies.

Load-bearing premise

The linear relationship between the indices and π-electronic energies remains stable and generalizable beyond the specific benzenoid structures and data points used in the regression fit.

What would settle it

A regression on a fresh set of benzenoid hydrocarbons yielding a correlation coefficient below 0.999 for either index would falsify the claim of exceptional reliability as predictors.

read the original abstract

In this work, the exponential sum-connectivity Gourava index ($e^{SGO(G)}$) and the exponential product-connectivity Gourava index ($e^{PGO(G)}$) are computed and comparatively analyzed for benzenoid hydrocarbons. Our results demonstrate that these descriptors exhibit a strong mutual correlation and provide enhanced sensitivity in modeling the structural characteristics of molecular graphs. Regression analysis reveals that both indices are exceptionally reliable predictors of $\pi$-electronic energies, achieving correlation coefficients exceeding $0.999$. Notably, a comparative assessment indicates that the exponential product-connectivity variant offers a slightly superior fit, as its coefficients align more precisely with optimal least-squares results. These findings confirm that both exponential Gourava-based indices provide a robust framework for characterizing electronic properties, with the product-connectivity version showing particular promise for high-precision QSPR studies in benzenoid systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper computes the exponential sum-connectivity Gourava index (e^{SGO(G)}) and exponential product-connectivity Gourava index (e^{PGO(G)}) for benzenoid hydrocarbons. Regression analysis is used to relate these indices to π-electronic energies, reporting correlation coefficients exceeding 0.999 and concluding that both are exceptionally reliable predictors, with the product-connectivity variant offering a slightly superior fit.

Significance. If the reported correlations are shown to be robust and generalizable, the indices could provide additional topological descriptors for QSPR modeling of electronic properties in benzenoid systems. The work supplies explicit computations of the new exponential variants and a direct comparison of their performance against the same property.

major comments (2)
  1. [Abstract] Abstract: The central claim that the indices are 'exceptionally reliable predictors' of π-electronic energies rests on regression coefficients exceeding 0.999, yet the manuscript supplies no count of molecules in the dataset, no cross-validation procedure, no error bars, and no statement that the fit was evaluated on held-out structures. Without these, the result demonstrates in-sample correlation rather than predictive reliability.
  2. [Abstract] Abstract and regression analysis: The paper describes the high R values as arising from least-squares fitting to the same π-electronic energy values being modeled; this renders the 'prediction' language circular, as the coefficients are optimized precisely to minimize residuals on the training molecules.
minor comments (1)
  1. [Abstract] The abstract states that the product-connectivity variant 'aligns more precisely with optimal least-squares results,' but this is expected by construction for any linear fit and does not constitute an independent advantage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract and regression presentation. The feedback correctly identifies that our original wording overstated the results as predictive without supporting validation details. We have revised the manuscript to use more precise language describing in-sample correlations and have added dataset information where feasible.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the indices are 'exceptionally reliable predictors' of π-electronic energies rests on regression coefficients exceeding 0.999, yet the manuscript supplies no count of molecules in the dataset, no cross-validation procedure, no error bars, and no statement that the fit was evaluated on held-out structures. Without these, the result demonstrates in-sample correlation rather than predictive reliability.

    Authors: We agree that the abstract's use of 'exceptionally reliable predictors' is not supported by out-of-sample validation or error analysis. The reported R values > 0.999 are from ordinary least-squares fits performed on the full set of benzenoid hydrocarbons for which the indices were computed. We have revised the abstract to state that both indices 'exhibit strong linear correlations' with the π-electronic energies (R > 0.999) rather than claiming predictive reliability. The number of structures examined has been added to the abstract and methods section. Because the work is a comparative index study rather than a QSPR model development paper, formal cross-validation was not included; the revision removes any implication of general predictive power. revision: yes

  2. Referee: [Abstract] The paper describes the high R values as arising from least-squares fitting to the same π-electronic energy values being modeled; this renders the 'prediction' language circular, as the coefficients are optimized precisely to minimize residuals on the training molecules.

    Authors: We concur that describing the indices as 'predictors' is inappropriate when the regression coefficients are fitted directly to the same property values. The high R values simply quantify the quality of the linear relationship within the studied molecules. The abstract has been rewritten to replace 'reliable predictors' and 'prediction' with 'strong correlation' and 'good fit,' making clear that the results are descriptive of the in-sample relationship. No claim of out-of-sample prediction is retained. revision: yes

Circularity Check

1 steps flagged

In-sample least-squares fit to π-energies labeled as 'reliable predictors' (R>0.999) without hold-out or cross-validation

specific steps
  1. fitted input called prediction [Abstract]
    "Regression analysis reveals that both indices are exceptionally reliable predictors of π-electronic energies, achieving correlation coefficients exceeding 0.999. Notably, a comparative assessment indicates that the exponential product-connectivity variant offers a slightly superior fit, as its coefficients align more precisely with optimal least-squares results."

    The quoted claim treats the output of least-squares regression (coefficients and R values) on the identical collection of benzenoid hydrocarbons as evidence that the indices 'predict' the π-energies. Because the fit is performed on the target values themselves, the reported correlation is the direct numerical consequence of the regression step rather than an independent verification of predictive power.

full rationale

The paper's central claim is that the exponential Gourava indices are 'exceptionally reliable predictors' of π-electronic energies on the basis of regression analysis yielding R>0.999. This reduces directly to ordinary least-squares performed on the same finite set of benzenoid structures used to compute both the indices and the energies; no independent test set, leave-one-out, or external validation is described. The abstract explicitly equates the fitted correlation with predictive reliability, matching the 'fitted input called prediction' pattern. No other derivation steps or self-citation chains are load-bearing.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the graph representations being faithful, the regression being the appropriate model, and the fitted coefficients generalizing; these are not derived from first principles but introduced to enable the reported correlations.

free parameters (1)
  • regression slope and intercept for each index
    Obtained by least-squares fit to the π-electronic energies of the studied molecules; the claim of superior fit for the product index rests on these fitted values.
axioms (1)
  • domain assumption The chosen benzenoid structures are representative and the graph-theoretic definitions of the indices correctly capture the relevant connectivity.
    Invoked when the indices are computed and then regressed against energies; no independent verification is described.

pith-pipeline@v0.9.1-grok · 5698 in / 1489 out tokens · 24427 ms · 2026-06-27T23:25:27.720525+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

10 extracted references · 4 canonical work pages

  1. [1]

    V. R. Kulli, The Product Connectivity Gourava Index, Journal of Computer and Mathematical Sciences.8(6)(2017) 235-242

    2017

  2. [2]

    V. R. Kulli, On the Sum Connectivity Gourava Index, International Journal of Mathematical Archive.8(6)(2017) 211-217

    2017

  3. [3]

    Sarwar, S

    K. Sarwar, S. Kanwal, A. Razzaque, QSPR analysis of amino acids for the family of Gourava indices, PLOS ONE.20(4)(2025). https://doi.org/10.1371/journal.pone.0319029

    work page doi:10.1371/journal.pone.0319029 2025

  4. [4]

    Dehmer, M

    M. Dehmer, M. Grabner, The discrimination power of molecular identification numbers revisited, MATCH Commun. Math. Comput. Chem.69(2013) 785–794

    2013

  5. [5]

    Rada, Exponential vertex-degree-based topological indices and discrimination, MATCH Commun

    J. Rada, Exponential vertex-degree-based topological indices and discrimination, MATCH Commun. Math. Comput. Chem.82(2019) 29–41

    2019

  6. [6]

    Carballosa, Y

    W. Carballosa, Y. Quintana, Jos´ e M. Rodr´ ıguez, Jos´ e M. Sigarreta, Exponential topological indices: optimal inequalities and applications, Journal of Mathematical Chemistry.(61)(2023) 933–949. https://doi.org/10.1007/s10910-022-01446-4. 11

    work page doi:10.1007/s10910-022-01446-4 2023

  7. [7]

    Hayat, S

    S. Hayat, S. J. F. Alanazi, M. B. Belay, S. Wang, Predictive power of para- metric temperature-based topological indices with applications in structure– property modeling of anti-tuberculosis drugs, Scientific Reports.15(2025) 39828. https://doi.org/10.1038/s41598-025-23529-3

    work page doi:10.1038/s41598-025-23529-3 2025

  8. [8]

    Luˇ ci´ c, N

    B. Luˇ ci´ c, N. Trinajsti´ c, B. Zhou, Comparison between the sum-connectivity index and product-connectivity index for benzenoid hydrocarbons, Chemical Physics Letters.475(2009) 146–148. https://doi.org/10.1016/j.cplett.2009.05.022

    work page doi:10.1016/j.cplett.2009.05.022 2009

  9. [9]

    Gutman, S

    I. Gutman, S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer, Berlin, (1989)

    1989

  10. [10]

    J. Rada, O. Araujo, I. Gutman, Randi´ c index of benzenoid systems and phenylenes, Croat. Chem. Acta.74(2)(2001) 225–235. 12

    2001