Optimizing predictive models for evaluating the F-temperature index in predicting the π-electron energy of polycyclic hydrocarbons, applicable to carbon nanocones

In the fields of mathematics, chemistry, and the physical sciences, graph theory plays a substantial role. Using modern mathematical techniques, quantitative structure-property relationship (QSPR) modeling predicts the physical, synthetic, and natural properties of substances based only on their chemical composition. For a chemical graph, the temperature of a vertex is a local property introduced by Fajtlowicz (1988). A temperature-based graphical descriptor is structured based on temperatures of vertices. Involving a non-zero real parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> , the general F-temperature index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>β</mml:mi></mml:msub> </mml:math> is a temperature index having strong efficacy. In this paper, we employ discrete optimization and regression analysis to find optimal value(s) of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> for which the prediction potential of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>β</mml:mi></mml:msub> </mml:math> and the total <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> -electron energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>π</mml:mi></mml:msub> </mml:math> of polycyclic hydrocarbons is the strongest. This, in turn, answers an open problem proposed by Hayat & Liu (2024). Applications of the optimal values for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>β</mml:mi></mml:msub> </mml:math> are presented a two-parametric family of carbon nanocones in predicting their <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi> <mml:mi>π</mml:mi></mml:msub> </mml:math> with significantly higher accuracy.