Faber-Krahn type inequality for supertrees

TL;DR

This paper extends the Faber-Krahn inequality to $k$-uniform supertrees with boundary, introducing a new ordering to identify extremal supertrees with minimal first eigenvalue for given degree sequences.

Contribution

It introduces the SLO-ordering for supertrees and characterizes the supertrees with the Faber-Krahn property among all with a given degree sequence.

Findings

SLO-supertree has the Faber-Krahn property among all supertrees with the same degree sequence.

The SLO-supertree with degree sequence $(d, dots,d, d', 1, dots, 1)$ has the Faber-Krahn property for degree sequences with minimum degree $d$.

The paper generalizes the Faber-Krahn inequality to a new class of hypergraph structures.

Abstract

The Faber-Krahn inequality states that the first Dirichlet eigenvalue among all bounded domains is no less than a Euclidean ball with the same volume in $\mathbb{R}^n$ \cite{Chavel FB}. B{\i}y{\i}ko\u{g}lu and Leydold (J. Comb. Theory, Ser. B., 2007) demonstrated that the Faber-Krahn inequality also holds for the class of trees with boundary with the same degree sequence and characterized the unique extremal tree. B{\i}y{\i}ko\u{g}lu and Leydold (2007) also posed a question as follows: Give a characterization of all graphs in a given class $\mathcal{C}$ with the Faber-Krahn property. In this paper, we address this question specifically for $k$-uniform supertrees with boundary. We introduce a spiral-like ordering (SLO-ordering) of vertices for supertrees, an extension of the SLO-ordering for trees initially proposed by Pruss [ Duke Math. J., 1998], and prove that the SLO-supertree has the Faber-Krahn property among all supertrees with a given degree sequence. Furthermore, among degree sequences that have a minimum degree $d$ for interior vertices, the SLO-supertree with degree sequence $(d,\ldots,d, d', 1, \dots, 1)$ possesses the Faber-Krahn property.