Walks and Paths in Trees
Bela Bollobas, Mykhaylo Tyomkyn

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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Stochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods
