Klobürštel Theorem on Trees

• The Klobürštel theorem is a discrete analogue of the classical Faber–Krahn theorem that defines a sharp spectral bound for finite trees based on interior and boundary vertex counts.
• It uniquely identifies a comet structure as the minimizer of the first Dirichlet eigenvalue, computed through Rayleigh quotient analysis and combinatorial Laplacians.
• The proof employs rearrangement moves and monotonicity principles, offering actionable insights for optimizing tree configurations in spectral graph theory.

The Klobürštel theorem is a discrete analogue of the classical Faber–Krahn theorem for the first Dirichlet eigenvalue of trees, providing a sharp spectral inequality for finite simple trees with given interior and boundary vertex counts. It establishes that among all such trees, a specific "comet" structure uniquely minimizes the first Dirichlet eigenvalue, providing both a precise inequality and a constructive characterization for extremal cases. The theorem situates itself in spectral graph theory and features rigorously articulated supporting concepts, including the combinatorial Laplacian, Dirichlet boundary conditions, and Rayleigh quotient analysis. Its development is attributed to the work of Bıyıkoğlu, Leydold, Lin, Liu, You, and Zhao, and incorporates earlier discrete Faber–Krahn results for regular trees and trees with fixed matching number (Lin et al., 5 Jan 2026).

1. Formal Definitions and Spectral Setup

Let $G = (V, E)$ denote a finite simple graph. A nonempty, proper subset $B \subsetneq V$ is designated the boundary. The interior, denoted $\Omega = V \setminus B$, is assumed to induce a connected subgraph. The graph's combinatorial Laplacian $\Delta$ acts on functions $f : V \to \mathbb{R}$ by

$$(\Delta f)(x) = \sum_{y \sim x} (f(y) - f(x)),\quad x \in V.$$

The Dirichlet eigenvalue problem in this context seeks solutions to

$$- \Delta f(x) = \lambda f(x), \quad x \in \Omega, \quad f|_B = 0,$$

whose spectrum $0 < \lambda_1 \le \lambda_2 \le \cdots$ is governed via the Rayleigh quotient

$$R_{(G, B)}(f) = \frac{\sum_{(x, y) \in E} (f(x) - f(y))^2}{\sum_{x \in \Omega} f(x)^2},$$

with

$\lambda_1 = \min_{f \not\equiv 0} R_{(G, B)}(f).$

A "tree with boundary" refers to a pair $T = (V, E; B)$ where $(V, E)$ is a tree and $B$ specifies the boundary. The matching number of $G$ is the largest cardinality of a set of pairwise disjoint edges.

2. Statement of the Theorem and Extremal Construction

For integers $n \ge 2$ and $1 \le k \le n - 1$, define

$$\mathcal T^{(n, k)} = \left\{ T :~ T\text{ is a tree on } n\text{ vertices},~ B = \{\text{leaves}\},~ |\Omega| = k \right\}$$

as the class of trees with $k$ interior vertices and boundary $B$ being the set of leaves. The "comet" $C_{n, k}$ is constructed by forming a path $x_0 \sim x_1 \sim \cdots \sim x_k$, declaring $x_0$ and all new leaves as boundary, and attaching $n - (k + 1)$ leaves at $x_k$. Thus, $\Omega = \{x_1, \dots, x_k\}$, $|B| = n - k$.

Klobürštel theorem: In the class $\mathcal T^{(n, k)}$, the first Dirichlet eigenvalue obeys the sharp bound

$$\lambda_1(T) \ge \lambda_1(C_{n, k}) = 2 \left(1 - \cos \frac{\pi}{k + 1}\right)$$

for every $T \in \mathcal T^{(n, k)}$, with equality if and only if $T$ is isomorphic to the comet $C_{n, k}$ (Lin et al., 5 Jan 2026).

3. Proof Ingredients and Monotonicity Principles

The proof consists of several combinatorial and variational arguments:

• Monotonicity under extensions (subgraphs): If $T' \subseteq T$ is obtained by converting interior vertices to boundary or deleting edges between boundary vertices (without increasing interior degrees), then

$\lambda_1(T) \le \lambda_1(T').$

Therefore, adding leaves or contracting interior edges can only raise $\lambda_1$.

• Rearrangement moves: Within a fixed combinatorial class, three elementary transformations do not increase the Rayleigh quotient for positive test functions:
  • Switching: Edges $v_1u_1$ and $v_2u_2$ are replaced by $v_1v_2$ and $u_1u_2$ when eigenfunction values are appropriately ordered.
  • Shifting: A pendant edge $uv_1$ is reattached to a deeper interior vertex $v_2$ if $f(v_1) \ge f(v_2) \ge f(u)$.
  • Jumping: When $u \sim v_1$ is on the geodesic from $v_1$ to $v_2$, one may replace $uv_1$ by $v_1v_2$ if $f(v_1) \ge f(v_2) \ge f(u)$.

Iterative application of these moves transforms any tree in $\mathcal T^{(n, k)}$—without increasing $\lambda_1$—to the comet $C_{n, k}$. The strictness of inequalities ensures the uniqueness of the minimizer.

4. Explicit Spectral Formula for the Comet Structure

For the comet $C_{n, k}$, the Dirichlet boundary is $\{x_0\}$ plus the $n-k-1$ leaves at $x_k$. Any Dirichlet eigenfunction must vanish at these. Therefore, the nonzero segment of the eigenfunction resides on the path

$$x_0 (=0) \sim x_1 \sim x_2 \sim \cdots \sim x_{k-1} \sim x_k (=0).$$

This path graph admits the Dirichlet spectrum

$$\lambda_1(C_{n, k}) = 2 \left(1 - \cos \frac{\pi}{k + 1}\right).$$

Thus, for any $T \in \mathcal T^{(n, k)}$,

$$\lambda_1(T) \ge 2 \left(1 - \cos \frac{\pi}{k + 1}\right),$$

with equality if and only if $T \cong C_{n, k}$.

5. Associated Formulas and Computational Perspectives

Key analytical expressions utilized include:

Quantity	Formula (LaTeX)	Description
Graph Laplacian	$\Delta f(x) = \sum_{y \sim x} (f(y) - f(x))$	Linear operator on vertex functions
Dirichlet Rayleigh	$$R_{(G,B)}(f) = \dfrac{\sum_{(x,y)\in E}(f(x)-f(y))^2}{\sum_{x\in\Omega}f(x)^2}$$	Spectral quotient on $(G,B)$
Faber–Krahn inequality	$$\lambda_1(T) \ge \lambda_1(C_{n,k}) = 2(1-\cos(\pi/(k+1)))$$	Main theorem

These formulas enable exact and numerically tractable eigenvalue bounds within the stated combinatorial constraints.

6. Sharpness Criteria and Extremal Examples

The sharpness of the Klobürštel theorem is exemplified by specific cases:

• For $k = 1$: $\mathcal T^{(n,1)}$ consists solely of the star $K_{1, n-1}$, whose first Dirichlet eigenvalue evaluates as $2$.
• For $k = n - 2$: The boundary has cardinality $2$; the unique extremal "comet" is the path $P_n$, with first Dirichlet eigenvalue $2(1 - \cos(\pi/(n-1)))$.
• Intermediate $k$ values: In each instance, the unique eigenvalue minimizer is the comet $C_{n, k}$.

A plausible implication is that structural optimizations of trees in spectral graph applications can be reduced to identifying isomorphisms with comet configurations for given interior-boundary counts.

7. Historical Context and Related Results

The continuum Faber–Krahn theorem posits that balls minimize the first Dirichlet eigenvalue among domains of fixed volume in $\mathbb{R}^n$. Discrete analogues for trees were introduced by Leydold (Geom. Funct. Anal., 1997) and generalized by Bıyıkoğlu–Leydold (J. Combin. Theory Ser. B, 2007), addressing degree sequences and matching numbers. Lin, Liu, You, and Zhao further established the Faber–Krahn property for trees with fixed matching number, and formulated the Klobürštel theorem as a corollary for trees with specified interior and boundary vertex counts (Lin et al., 5 Jan 2026).

This body of research provides rigorous combinatorial and spectral tools for characterizing optimal tree structures under Dirichlet constraints, serving as foundational references for ongoing work in spectral graph theory and combinatorial optimization.