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arXiv · 2601.01859

A Faber--Krahn inequality for trees

Huiqiu Lin·Lianping Liu·Zhe You
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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 $\mathbb{R}^n$. 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 \(\mathcal{C}\) 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.

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Huiqiu Lin, Lianping Liu, Zhe You. 2026-01-05. A Faber--Krahn inequality for trees. https://arxiv.org/abs/2601.01859

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