LAUSR

For a graph G, and two distinct vertices u and v of G, let nG(u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{G}(u,v)$$\end{document} be the number of vertices of G that are closer in G to u than to v. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness of G as the sum of |nG(u,v)-nG(v,u)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|n_{G}(u,v)-n_{G}(v,u)|$$\end{document} over all unordered pairs of distinct vertices u and v of G. Confirming one of their conjectures, we show that the stars minimize the distance-unbalancedness among all trees of a fixed order.