Lower bounds on trees and unicyclic graphs with respect to the misbalance rodeg index

The Misbalance Rodeg (MR) index stands out among the 148 discrete Adriatic indices demonstrating considerable predictive capabilities in evaluations carried out by the International Academy of Mathematical Chemistry. This index excels particularly in forecasting both the enthalpy and the standard enthalpy of vaporization for octane isomers. Despite its significant chemical applicability, the MR index has not been extensively explored in the literature. One objective of this study is to highlight the importance of this graph invariant to the mathematical chemistry community by examining various mathematical properties associated with it. Our investigation specifically aims to ascertain the minimal values of the MR index for all trees and unicyclic graphs with a given order and maximum vertex degree. Additionally, we extend our analysis to molecular trees and provide a characterization of the respective minimal trees and unicyclic graphs.