A Faber--Krahn inequality for trees
Abstract
The well-known Faber-Krahn theorem states that the ball has the lowest first Dirichlet eigenvalue among all domains of the same volume in . Leydold (Geom. Funct. Anal, 1997) gave the discrete version of Faber-Krahn inequality for regular trees with boundary. B{\i}y{\i}ko{\u{g}}lu and Leydold (J. Combin. Theory Ser. B, 2007) demonstrated that the Faber--Krahn inequality holds for the class of trees with boundary with the same degree sequence. They further posed the following question: Give a characterization of all graphs in a given class with the Faber-Krahn property. In this paper, we show the Faber-Krahn property for trees with given matching number. Our result can imply the Klob\"ur\v{s}tel theorem, i.e., the Faber-Krahn inequality for trees with given number of interior vertices and boundary vertices.
Keywords
Cite
@article{arxiv.2601.01859,
title = {A Faber--Krahn inequality for trees},
author = {Huiqiu Lin and Lianping Liu and Zhe You},
journal= {arXiv preprint arXiv:2601.01859},
year = {2026}
}