On the Szeged–Sombor Index of Graphs

Abstract: In this paper, we propose a geometrical interpretation of bond additive indices, especially Szeged type indices. Building on this, we introduce a new class of bond additive indices, namely the Szeged–Sombor index, and study its properties. We determine the Szeged–Sombor index for elementary graphs and determine its relationship with other topological indices. Additionally, we derive an explicit expression for the Szeged–Sombor index of the Cartesian product of graphs. We then establish both upper and lower bounds for the Szeged–Sombor index of trees and bipartite graphs in terms of their order, and characterize the graphs that attain these bounds. Additionally, we provide an upper bound for the Szeged–Sombor index of unicyclic graphs with a fixed order, and identify the extremal graphs. We also present an upper bound on the Szeged–Sombor index of a graph in terms of, and, and characterize the extremal graphs. We conclude our study by discussing the chemical significance of Szeged–Sombor index by analyzing its values on octane isomers and benzenoid hydrocarbons. We demonstrate that the newly proposed Szeged–Sombor index shows a significantly higher correlation with the chemical properties of the compounds compared to some other distance-based topological indices. Finally, we propose some open problems for future research.