On the Adjacency Spectra of Hypertrees

TL;DR

This paper explores the spectral properties of k-uniform hypertrees, linking eigenvalues to matching polynomials and providing characterizations that differ from ordinary trees, with implications for understanding hypertree structures.

Contribution

It extends spectral results to hypertrees, establishing a connection between eigenvalues and matching polynomials, and characterizes power hypertrees uniquely.

Findings

Eigenvalues correspond to roots of matching polynomials for connected subtrees.

Spectral characterization of power hypertrees differs from ordinary trees.

Example of a non-power hypertree illustrating these spectral phenomena.

Abstract

We extend the results of Zhang et al. to show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., $2$-uniform trees. We conclude by presenting an example (an $11$ vertex, $3$-uniform non-power hypertree) illustrating these phenomena.