CMMSE: Upper and Lower Bounds for the Sum-Lordeg Index on unicyclic graphs

Topological indices are used to understand physicochemical properties of chemical compounds,since they capture some properties of a molecule in a single number. The \emph{sum lordeg index} is defined as$$SL(G) = \sum_{u\in V(G)} d_u \sqrt{\log d_u} \,.$$This index is interesting from an applied viewpoint sinceit is the best predictor of octanol-water partitioncoefficient for octane isomers. The aim of this paper is to obtain new inequalities for the Sum-Lordeg Index restricted to unicyclic graphsand to characterize the set of extremal graphs.Our main results provide upper and lower bounds for this topological index on unicyclic graphs, fixing or not the number of pendant vertices, and finding the corresponding extremal graphs.Also, we provide some new relations between the Sum-Lordeg Index and other indices as the First Variable Zagreb index or the Narumi-Katayama Index.Finally, we show that the sum lordeg index is an important tool for predicting the boiling point of cycloalkanes isomers. MSC Classi cation: 05C09 , 05C92 , 92E10.