Mathematical and chemical properties of geometry-based invariants and its applications

Recently, based on elementary geometry, Gutman proposed several geometry-based invariants (i.e., $SO$, $SO_{1}$, $SO_{2}$, $SO_{3}$, $SO_{4}$, $SO_{5}$, $SO_{6}$). The Sombor index was defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, the first $SO_{1}(G)= \frac{1}{2}\sum\limits_{uv\in E(G)}|d_{u}^{2}-d_{v}^{2}|$, where $d_{u}$ denotes degree of vertex $u$. In this paper, we consider mathematical and chemistry properties these invariants. We determine maximum trees (resp. unicyclic graphs) with given diameter, matching number, pendent vertices, minimum trees) branching second degree, graphs girth, connected some vertices respect to $SO_{1}$. As an application, inaugurate verify their chemical applicability. used model acentric factor entropy, enthalpy vaporization, etc.) alkanes, obtained satisfactory predictive potential, which indicates that can be successfully thermodynamic compounds.