A Faber--Krahn inequality for trees

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.