Complexity measures in trees: a comparative investigation of Szeged and Wiener indices

The study of graph complexity has led to a deeper understanding of the structures of graphs. This paper presents new findings on the Szeged complexity of graphs. Specifically, we prove that for bipartite graphs on n vertices, the upper bound of Szeged complexity is & LeftFloor;n2 & RightFloor;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor \frac{n}{2} \rfloor $$\end{document}. Moreover, we establish that the lower bound of Szeged complexity of a tree T is the radius of T. Furthermore, we characterize trees with Szeged complexity three and determine their Wiener complexity.