Minimum distance-unbalancedness of trees

TL;DR

This paper proves that among all trees of a fixed size, stars have the minimum distance-unbalancedness, confirming a conjecture related to graph symmetry and vertex distance distributions.

Contribution

The paper confirms a conjecture that stars minimize the distance-unbalancedness among all trees of a given order.

Findings

Stars minimize the distance-unbalancedness among all trees of fixed order.

The result confirms a previously conjectured property of trees.

The study advances understanding of graph symmetry and distance properties.

Abstract

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ 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.