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A Faber--Krahn inequality for trees

Huiqiu Lin, Lianping Liu, Zhe You

TL;DR

This paper extends Faber–Krahn type questions to discrete graphs, focusing on trees with boundary and fixed matching number or specified interior/boundary counts. Using variational characterizations of the first Dirichlet eigenvalue and a toolkit of edge-rearrangement operations (switching, shifting, jumping), the authors prove precise extremal classifications in two classes: $\mathcal{T}(n,m)$ and $\mathcal{T}(n,m,b)$. For $\mathcal{T}(n,m)$, the extremal structures are comet-like ball approximations, with a key outcome that $\lambda_1$ is minimized by $T(2m-3,2,n+1-2m)$. In the broader class $\mathcal{T}(n,m,b)$, the extremals are governed by $t=2m+b-n$, yielding explicit isomorphism types and special cases where all interior vertices meet particular leaf-adjacency conditions. The work further connects to Klobürštel's theorem and extends considerations to diameter-constrained families via generalized fork graphs, suggesting a path toward extremal results for planar graphs and maximal planar graphs with boundary.

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. 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.

A Faber--Krahn inequality for trees

TL;DR

This paper extends Faber–Krahn type questions to discrete graphs, focusing on trees with boundary and fixed matching number or specified interior/boundary counts. Using variational characterizations of the first Dirichlet eigenvalue and a toolkit of edge-rearrangement operations (switching, shifting, jumping), the authors prove precise extremal classifications in two classes: and . For , the extremal structures are comet-like ball approximations, with a key outcome that is minimized by . In the broader class , the extremals are governed by , yielding explicit isomorphism types and special cases where all interior vertices meet particular leaf-adjacency conditions. The work further connects to Klobürštel's theorem and extends considerations to diameter-constrained families via generalized fork graphs, suggesting a path toward extremal results for planar graphs and maximal planar graphs with boundary.

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. 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 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.
Paper Structure (6 sections, 17 theorems, 43 equations, 7 figures)

This paper contains 6 sections, 17 theorems, 43 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs
  4. Faber–Krahn property in T(n,m)
  5. Faber–Krahn property in T(n,m, b)
  6. Concluding remarks

Key Result

Theorem 1.2

A tree $T$ has the Faber-Krahn property in the class $\mathcal{T}^{(n,k)}$ if and only if $T$ is a star with a long tail, i.e., a comet, see a comet. $T$ is then uniquely determined up to isomorphism.

Figures (7)

  • Figure 1: A comet has the Faber-Krahn property in class $\mathcal{T}^{(n,k)}$.
  • Figure 2: The tree $T(p, q, b)$
  • Figure 3: Switching : $v_1u_1$ and $v_2u_2$ are replaced by $v_1v_2$ and $u_1u_2$.
  • Figure 4: Shifting: $uv_1$ is replaced by $uv_2$.
  • Figure 5: Jumping: $uv_1$ is replaced by $uv_2$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1.2: Klobürštel theorem biyikouglu2007faber
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1: lin2025estimatesdirichleteigenvaluegraphs
  • Theorem 2.2
  • Lemma 2.3: Switching biyikouglu2007faber
  • Lemma 2.4: biyikouglu2007faber
  • Lemma 2.5: Shifting biyikouglu2007faber
  • Lemma 2.6: Jumping
  • ...and 25 more