The spectrum of a class of uniform hypergraphs

TL;DR

This paper explores the spectral properties of generalized power hypergraphs, showing how their eigenvalues relate to those of their base hypergraphs and subgraphs, with new results on edge-expansion operations.

Contribution

It provides a method to compute all nonzero eigenvalues of generalized power hypergraphs from their base hypergraphs and subgraphs, advancing spectral hypergraph theory.

Findings

Eigenvalues of hypergraphs can be derived from base hypergraphs and subgraphs.

Spectral results for edge-expansion operations are established.

All nonzero eigenvalues of generalized power hypergraphs are characterized.

Abstract

A generalized power hypergraph $\mathcal{H}^k_s$ is obtained from a base hypergraph $\mathcal{H}$ by means of some simple edge-expansion operations. Kang, Liu, Qi and Yuan [8] proved that the nonzero eigenvalues of $\mathcal{H}$ give rise to nonzero eigenvalues of $\mathcal{H}^k_s$. In this paper we show that all nonzero eigenvalues of $\mathcal{H}^k_s$ may be computed from the eigenvalues of its base hypergraph $\mathcal{H}$ and of its subgraphs. To prove this, we derive spectral results about edge-expansion operations that may be interesting on their own sake.