Faber-Krahn Trees: The Comet Minimizer

• Faber-Krahn property for trees is defined by minimizing the first Dirichlet eigenvalue among trees with fixed numbers of interior and boundary vertices, analogous to classical domains.
• The comet configuration—a star augmented with a path—is proven to be the unique minimizer within its class, as established by the Klobürštel theorem.
• The proof employs combinatorial rearrangement techniques (switching, shifting, and jumping) to iteratively transform any tree into the optimal comet structure.

The Faber-Krahn property for trees concerns the minimization of the first Dirichlet eigenvalue of the graph Laplacian over all trees within a class specified by structural constraints, such as order, boundary configuration, or matching number. In direct analogy to the classical Faber-Krahn theorem in $\mathbb{R}^n$, which asserts that the Euclidean ball minimizes the first Dirichlet eigenvalue among domains of fixed volume, the discrete setting seeks extremal configurations among graphs—particularly trees—with prescribed combinatorial parameters. Recent developments provide a full characterization for trees with fixed numbers of interior and boundary vertices, notably via the Klobürštel theorem, which establishes the unique minimizer within this structural class as a specific construction called the "comet" (Lin et al., 5 Jan 2026).

1. Terminology and Setup

Let $T = (V, E; B)$ denote a tree with vertex set $V$, edge set $E$, and boundary (i.e., set of boundary vertices) $B \subset V$. The interior is $\Omega = V \setminus B$. The combinatorial Laplacian $\Delta$ acts on functions $f : V \to \mathbb{R}$ by

$(\Delta f)(x) = \sum_{y \sim x} (f(x) - f(y)).$

The Dirichlet eigenvalue problem on $(T,B)$ requires $f(x) = 0$ for $x \in B$ and $-\Delta f(x) = \lambda f(x)$ for $x \in \Omega$, yielding eigenvalues

$$0 < \lambda_1(T,B) \leq \lambda_2(T,B) \leq \cdots \leq \lambda_{|\Omega|}(T,B).$$

The Rayleigh principle gives

$$\lambda_1(T,B) = \min_{0 \neq f \in \mathbb{R}^\Omega} \frac{\sum_{(x,y) \in E} (f(x) - f(y))^2}{\sum_{x \in \Omega} f(x)^2}.$$

The Faber-Krahn property for a class $\mathcal{C}$ is defined: a graph $G \in \mathcal{C}$ has the property if it minimizes $\lambda_1$ among all graphs in $\mathcal{C}$ with a fixed notion of "volume," here interpreted as prescribed $|\Omega|$ and $|B|$.

2. The Klobürštel Theorem and Characterization

Consider $$\mathcal{T}^{(n,k)} = \{ T=(V,E;B) : |V| = n, ~ |\Omega| = k, ~ B = \{\text{leaves of }T\} \}$$. The Klobürštel theorem asserts: $T \in \mathcal{T}^{(n,k)}$ attains the Faber-Krahn property precisely if $T$ is a "comet"—a tree where a star $K_{1, n-k}$ is augmented by attaching a simple path of length $k-1$ to the center. Formally, $T = K_{1, n-k} \cup \text{path}_{k-1}$ glued at the star center. The leaves of $K_{1, n-k}$ and the terminal of the path comprise $B$; the $k$ interior vertices are the path vertices and the center. The comet is unique up to isomorphism for $(n,k)$ (Lin et al., 5 Jan 2026).

3. Spectral Properties and Domain Monotonicity

The first Dirichlet eigenvalue on a connected graph with boundary is simple, with a strictly positive eigenfunction on the interior and vanishing on the boundary. Further, domain monotonicity holds: if interior vertices are declared boundary, or boundary edges are shortened to vertices, then every eigenvalue increases, i.e.,

$\lambda_k(G') \geq \lambda_k(G)$

for the resulting graph $G'$. For trees with fixed interior and boundary sets, ball-like (radially symmetric) configurations minimize $\lambda_1$; in this context, the comet serves as the extremal minimizer.

4. Combinatorial Rearrangement Lemmas

The proof exploits three tree surgery moves—Switching, Shifting, and Jumping—all of which preserve the degree sequence and boundary, while non-increasing the Rayleigh quotient for any fixed nonnegative test function. Taking the Dirichlet eigenfunction as the test function, strict inequalities in certain function values strictly reduce $\lambda_1$:

• Switching: Replace edges $v_1u_1, v_2u_2$ by $v_1v_2, u_1u_2$ when $u_1$ is off, and $u_2$ on, the path $v_1$–$v_2$.
• Shifting: For an edge $uv_1$, replace it with $uv_2$ if $u$ is not on the $v_1$–$v_2$ path.
• Jumping: For $uv_1$ lying on the $v_1$–$v_2$ geodesic with $u$ adjacent to a boundary, delete $uv_1$ and add $v_1v_2$.

At each such transformation, for properly ordered $f$ values, the Rayleigh quotient does not increase and strictly drops if strict inequalities occur. This iterative rearrangement approach converges to the comet shape.

5. Proof Structure of the Faber-Krahn Property for Trees

Given $T \in \mathcal{T}^{(n,k)}$ and its Dirichlet ground state $f > 0$, order the $k$ interior vertices as $v_1, ..., v_k$ with $f(v_1) \geq \cdots \geq f(v_k) > 0$. By successively applying the combinatorial rearrangement lemmas, $T$ is transformed so that $v_1$ connects directly to all other interior vertices and forms the center of the star, with the remainder forming the path. Each rearrangement either preserves or strictly lowers the Rayleigh quotient. For $T$ not already the comet, a strict drop occurs, establishing the comet as the unique minimizer for $\lambda_1$ in $\mathcal{T}^{(n,k)}$ (Lin et al., 5 Jan 2026).

6. Concrete Examples and Spectral Extremality

• For $k=1$, $\mathcal{T}^{(n,1)}$ is the set of stars $K_{1,n-1}$; the Faber-Krahn minimizer is the star itself.
• For $k=2$, the extremal is the comet with degree sequence $(n-2,2)$: all $n-2$ leaves are attached to the stronger of the two interior vertices.

Generally, the comet is interpreted as a high-degree center with a tail of length $k-1$. The first Dirichlet eigenfunction decays along this tail, and extremality follows from the ability to manipulate any tree in the class into comet form without increasing $\lambda_1$.

7. Implications and Further Directions

The discrete Faber-Krahn classification for trees with prescribed numbers of interior and boundary vertices resolves the conjecture raised by Bıyıkoğlu and Leydold in the context of trees with the same degree sequence, extending earlier results of Leydold for regular trees. The uniqueness and explicit construction of the Faber-Krahn minimizer as the comet highlights the sharp connection between spectral minimization and structural graph properties in combinatorial settings (Lin et al., 5 Jan 2026). The result also subsumes the Faber-Krahn property for trees with given matching number and reduces to classical results for edge-regular stars. Further directions include characterizations in wider graph classes and exact structural constraints for extremizers under alternative boundary or degree sequences.