A comparative study of sum-connectivity and product-connectivity Gourava indices for benzenoid hydrocarbons

Technical Summary: A Comparative Study of Sum-Connectivity and Product-Connectivity Gourava Indices for Benzenoid Hydrocarbons

Problem Statement
This study addresses the need to evaluate the efficacy of recently introduced topological descriptors—the sum-connectivity Gourava index ($SGO$) and the product-connectivity Gourava index ($PGO$)—in characterizing benzenoid hydrocarbons. While classical connectivity indices (such as the Randić index and its variants) have long been established for predicting physicochemical properties like $\pi$-electronic energy ($E_\pi$), the performance of the newer Gourava indices in this domain requires systematic validation. The central problem is to determine whether $SGO$ or $PGO$ offers superior predictive power and structural discrimination for aromatic systems, and to rigorously validate the more promising candidate against established metrics.

Methodology
The research employs a hierarchical validation framework involving 30 lower benzenoid hydrocarbons and additional datasets for broader structural testing:

1. Derivation and Calculation: The authors derived simplified formulas for $SGO$ and $PGO$ specific to benzenoid graphs, where vertices have degrees of 2 or 3. These formulas were expressed in terms of edge types ($e_{22}, e_{23}, e_{33}$) and further refined using structural parameters: the number of vertices ($n$), hexagons ($h$), and inlets ($r$).
2. Regression Analysis: A least-squares regression analysis was performed to correlate both indices with $\pi$-electronic energy ($E_\pi$) values obtained from literature. The study compared the regression coefficients of the Gourava indices against an "optimal" model derived directly from edge-type counts to assess predictive accuracy.
3. Intercorrelation Analysis: The $SGO$ index was compared against seven other prominent degree-based topological indices (First/Second Zagreb, Harmonic, Sombor, Diminished Sombor, Atom-Bond-Sum Connectivity, and Nirmala) to determine linear dependencies and unique structural information.
4. Degeneracy Analysis: The discriminatory power of $SGO$ was tested against the other indices using three datasets: Octane isomers ($N=18$), Nonane isomers ($N=35$), and all non-isomorphic trees of order 10 ($N=106$).
5. Structure-Sensitivity Analysis: The responsiveness of $SGO$ to minor structural changes was quantified using Structure Sensitivity ($SS$), Abruptness ($Abr$), and the Structure-Abruptness ratio ($SA$) across trees of order $n=10$.

Key Contributions and Results

• Superior Predictive Fit for $E_\pi$: The regression analysis revealed that $SGO$ yields a markedly better fit for predicting $\pi$-electronic energies than $PGO$.
  • $SGO$ achieved a correlation coefficient ($R$) of 0.9997 with $E_\pi$, compared to 0.9970 for $PGO$.
  • The regression coefficients derived from the $SGO$ model closely matched the optimal least-squares coefficients obtained from direct edge-type analysis, whereas the $PGO$ model showed significant deviation, particularly in the intercept and edge-type weighting.
• Structural Independence: Correlation analysis showed that while $SGO$ is strongly negatively correlated with indices like $M_1$, $M_2$, $SO$, $DSO$, $ABS$, and $N$ (coefficients ranging from $-0.8936$ to $-0.9923$), it does not exhibit a perfect inverse relationship ($|r| \neq 1$). This suggests $SGO$ captures unique structural variance not fully represented by these other descriptors.
• Low Degeneracy: In degeneracy testing, $SGO$ demonstrated high discriminatory power, matching the performance of modern indices like Sombor ($SO$) and Diminished Sombor ($DSO$).
  • Octane: $22.22\%$ degeneracy.
  • Nonane: $40.00\%$ degeneracy.
  • Order-10 Trees: $42.45\%$ degeneracy.
  • These values were significantly lower than classical indices like $M_1$ (up to $97.14\%$) and $M_2$.
• Sensitivity vs. Stability: In the structure-sensitivity analysis, $SGO$ achieved a Structure-Abruptness ratio ($SA$) of 0.474386, ranking third among the eight tested indices (behind Harmonic and $DSO$). Crucially, while $SGO$ sacrificed only a negligible $0.8\%$ in $SA$ compared to the top performer ($DSO$), it provided a 74% higher structure sensitivity ($0.034429$ vs. $0.019758$), indicating a superior ability to distinguish between similar molecular structures without sacrificing numerical stability.

Significance and Claims
The paper concludes that the sum-connectivity Gourava index ($SGO$) is a robust molecular descriptor that offers a favorable balance between discriminative power and numerical stability. The authors claim that $SGO$ is statistically superior to the product-connectivity variant ($PGO$) for predicting $\pi$-electronic energies in benzenoid systems. Furthermore, the study posits that $SGO$ provides complementary structural information to existing descriptors, evidenced by its non-perfect inverse correlations and low degeneracy rates.

The authors assert that these findings support the applicability of $SGO$ in Quantitative Structure-Property Relationship (QSPR) modeling and related theoretical studies, particularly where distinguishing between non-isomorphic structures is critical. The paper modestly suggests that these results warrant further investigation into the mathematical properties and broader chemical applications of the $SGO$ index, without proposing specific new experimental protocols or unverified applications.