Walks and Paths in Trees

TL;DR

This paper extends Csikvári's theorem from closed walks to all walks in trees, provides a simpler proof, addresses an open question negatively, and explores similar questions for paths.

Contribution

It generalizes existing results on walks in trees, offers a simplified proof, and investigates analogous path problems.

Findings

Extended Csikvári's theorem to all walks in trees.

Provided a simpler proof of Csikvári's result.

Answered one of Csikvári's open questions negatively.

Abstract

Recently Csikv\'ari \cite{csik} proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend his theorem to all walks. In addition, we give a simpler proof of Csikv\'ari's result and answer one of his questions in the negative. Finally we consider an analogous question for paths rather than walks.