Graphs with given connectivity and their minimum Sombor index having applications to QSPR studies of monocarboxylic acids

Abstract

The Sombor index, introduced by Gutman in 2021, represents a novel graphical topological descriptor reliant upon graph degree information. It holds promise for elucidating the thermodynamic behavior of compounds. Denoting by $V_n^k$ (respectively, $E_n^k$) the set encompassing all connected graphs comprising n vertices with a specific vertex-connectivity (correspondingly, edge-connectivity) value of k. In Problem 1 (resp. Problem 2) of Hayat et al. (2022) [33], the question of finding minimum Sombor index of graphs in $V_n^k$ (resp. $E_n^k$) is proposed. In this note, we solve both of the open problems by characterizing minimizing graphs in $E_n^k$ and $V_n^k$ corresponding to the Sombor index. We employed commonly occurring valency-based graphical indices in a QSPR studies of the physicochemical characteristics for monocarboxylic acids. The proposed statistical models infer that the Sombor index predicts various physicochemical characteristics for monocarboxylic acids having strong predictive coefficients such as $\rho =0.99998$.

Keywords: 05C09; 05C35; 05C92; Edge-connectivity; Extremal values; Graphs; QSPR model; Sombor index; Vertex-connectivity.