On majorization of closed walks vector of trees with given degree sequences

TL;DR

This paper proves that among trees with a fixed degree sequence, the greedy tree minimizes the vector of closed walk counts starting at each vertex, and that majorization of degree sequences leads to ordered closed walk vectors.

Contribution

It establishes a majorization relation for closed walks in trees with given degree sequences and identifies the greedy tree as extremal for these counts.

Findings

The sequence of closed walk counts is weakly majorized by that of the greedy tree.

Majorization of degree sequences implies ordered closed walk vectors for the corresponding greedy trees.

The results characterize extremal trees with respect to closed walk distributions.

Abstract

Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $\pi$, then for any tree with degree sequence $\pi$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_{\pi}^*)\equiv C(k, T_{\pi}^*, v\in V(T^*))$, where $T_{\pi}^*$ is the greedy tree corresponding to $\pi$. In addition, for two trees degree sequences $\pi,~\pi'$, if $\pi$ is majorized by $\pi'$, then $C(k;T_{\pi}^*)$ is weakly majorized by $C(k;T_{\pi'}^*)$.