Maximally distance-unbalanced trees

TL;DR

This paper investigates the maximum distance-unbalancedness of trees, establishing asymptotic bounds and characterizing trees that maximize this measure, especially subdivided stars, to advance understanding of graph imbalance metrics.

Contribution

The authors determine the asymptotic maximum of the distance-unbalancedness for trees and identify subdivided stars as the extremal structures for large n.

Findings

Maximum uB(T) is approximately n^3/2 for large n.

Maximum uB among subdivided stars depends on the partition of n into path lengths.

Subdivided stars are shown to asymptotically maximize the distance-unbalancedness.

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 ${\rm uB}(G)$ 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 $G$. For positive integers $n$ up to $15$, they determine the trees $T$ of fixed order $n$ with the smallest and the largest values of ${\rm uB}(T)$, respectively. While the smallest value is achieved by the star $K_{1,n-1}$ for these $n$, which we then proved for general $n$ (Minimum distance-unbalancedness of trees, Journal of Mathematical Chemistry, DOI 10.1007/s10910-021-01228-4), the structure of the trees maximizing the distance-unbalancedness remained unclear. For $n$ up to $15$ at least, all these trees were subdivided stars. Contributing to problems posed by Miklavi\v{c} and \v{S}parl, we show $$\max\Big\{{\rm uB}(T):T\mbox{ is a tree of order }n\Big\} =\frac{n^3}{2}+o(n^3)$$ and $$\max\Big\{{\rm uB}(S(n_1,\ldots,n_k)):1+n_1+\cdots+n_k=n\Big\} =\left(\frac{1}{2}-\frac{5}{6k}+\frac{1}{3k^2}\right)n^3+O(kn^2),$$ where $S(n_1,\ldots,n_k)$ is the subdivided star such that removing its center vertex leaves paths of orders $n_1,\ldots,n_k$.