A Faber--Krahn inequality for trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918367109054464 |
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| author | Lin, Huiqiu Liu, Lianping You, Zhe |
| author_facet | Lin, Huiqiu Liu, Lianping You, Zhe |
| contents | 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. Bıyıko{ğ}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 Klobürštel theorem, i.e., the Faber-Krahn inequality for trees with given number of interior vertices and boundary vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01859 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Faber--Krahn inequality for trees Lin, Huiqiu Liu, Lianping You, Zhe Combinatorics Differential Geometry 05C05, 05C50, 35J25, 53A70 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. Bıyıko{ğ}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 Klobürštel theorem, i.e., the Faber-Krahn inequality for trees with given number of interior vertices and boundary vertices. |
| title | A Faber--Krahn inequality for trees |
| topic | Combinatorics Differential Geometry 05C05, 05C50, 35J25, 53A70 |
| url | https://arxiv.org/abs/2601.01859 |