Sharp upper bound on the Sombor index of bipartite graphs with a given diameter

Let G be a connected graph. The Sombor index of a graph G is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(G)=\sum _{uv\in E(G)}\sqrt{d^2_{G}(u)+d^2_{G}(v)}$$\end{document}, where dG(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_G(u)$$\end{document} denotes the degree of u in G. Let Bnd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {B}}^d_n$$\end{document} be the set of all bipartite graphs of diameter d with n vertices. In this paper, we determine the sharp upper bound on the Sombor index of G∈Bnd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\in {\mathscr {B}}^d_n$$\end{document}. In addition, we propose an algorithm for searching the largest Sombor index among Bnd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {B}}^d_n$$\end{document}. Furthermore, the relationship between the maximal Sombor index of G∈Bnd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\in {\mathscr {B}}^d_n$$\end{document} and the diameter d is established. Finally, we obtain the largest, the second-largest, the third-largest and the smallest Sombor indices of bipartite graphs.