On the Szeged-Sombor Index of Graphs

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sz(G)$$\end{document}, 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.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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sz(G)$$\end{document}, 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.