A Faber--Krahn inequality for trees

TL;DR

This paper extends the Faber-Krahn inequality to trees with a specified matching number, providing a new characterization and implications for trees with fixed interior and boundary vertices.

Contribution

It establishes the Faber-Krahn property for trees with a given matching number, advancing the understanding of eigenvalue inequalities in discrete structures.

Findings

Faber-Krahn property proven for trees with specified matching number

Implication for Klobürstel theorem on trees with fixed interior and boundary vertices

Extension of Faber-Krahn inequality to new class of discrete graphs

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.