Trees, unicyclic graphs and their geometric Sombor index: an extremal approach

The Sombor index is a novel vertex-degree-based topological index which was proposed by Gutman in 2021. This index has been shown to be helpful in predicting the enthalpy of vaporization and entropy of octane isomers. In 2022, Gutman put forward a new variant of the Sombor index and gave its geometric interpretation. This invariant which we call the geometric Sombor index can be considered as a practical tool for measuring irregularity in graphs. Our aim is to study some basic mathematical properties of this new Sombor-type invariant. Especially, we prove that for any tree (resp. unicyclic graph) with a fixed order and maximum degree \(\Delta \) $\mathrm{\Delta }$ , the geometric Sombor index is bounded below by \(\frac{\Delta ^3-\Delta }{2}\) $\frac{{\mathrm{\Delta }}^3-\mathrm{\Delta }}{2}$ (resp. \(\frac{\Delta ^3-\Delta -6}{2}\) $\frac{{\mathrm{\Delta }}^3-\mathrm{\Delta }-6}{2}$ ). Also the extremal trees and unicyclic graphs that achieve the lower bound are characterized.