Comparative analysis of temperature-based graphical indices for correlating the total π-electron energy of benzenoid hydrocarbons

In a graph $G=(V,E)$, the temperature $T_x$ of a vertex $x\in V$ is defined as $T_x=d_x/n-d_x$, where n is the order of G and $d_x$ is the valency/degree of x. A topological/graphical index $\mathrm{\text{GI}}$ is a map $\mathrm{\text{GI}}:\sum \to ℝ$, where ∑ (respectively, $ℝ$) is the set of simple connected graphs (respectively, real numbers). Graphical indices are employed in quantitative structure-property relationship (QSPR) modeling to predict physicochemical/thermodynamic/biological characteristics of a compound. A temperature-based graphical index of a chemical graph G is defined as ${\mathrm{\text{GI}}}_T:={\sum }_{\mathrm{\text{edges}}}f(T_x,T_y)$, where $f(T_x,T_y)$ is a symmetric 2-variable map. In this paper, we introduce two new novel temperature-based indices named as the reduced reciprocal product-connectivity temperature ($\mathrm{\text{RRPT}}$) index and the geometric-arithmetic temperature ($\mathrm{\text{GAT}}$) index. The predictive potential of these indices has been investigated by employing them in structure-property modeling of the total $\pi$-electronic energy $E_{\pi }(\beta )$ of benzenoid hydrocarbons. In order to validate the statistical inference, the lower 30 BHs have been opted as test molecules as their experimental data for $E_{\pi }(\beta )$ is also publicly available. First, we employ a computer-based computational method to compute temperature indices of 30 lower BHs. Certain QPSR models are proposed by utilizing the experimental data of $E_{\pi }$ for the BHs. Our statistical analysis suggests that the most efficient regression models are, in fact, linear. Our statistical analysis asserts that both $\mathrm{\text{RRPT}}$ and $\mathrm{\text{GAT}}$ outperformed all the existing temperature indices for correlating $E_{\pi }$ for the BHs. The results suggest their further employability in QSPR modeling. Importantly, our research contributes toward countering proliferation of graphical indices.