Eigenvalues of Hypergraph Products and Reciprocal Eigenvalue Property

TL;DR

This paper explores how eigenvalues of hypergraph matrices behave under various hypergraph products, extending spectral symmetry concepts and revealing that power hypertrees lack the reciprocal eigenvalue property.

Contribution

It introduces the study of eigenvalues in hypergraph products and extends the reciprocal eigenvalue property to hypergraphs, providing new insights into hypergraph spectral theory.

Findings

Eigenvalues of hypergraph products are characterized under join, Kronecker, and corona operations.

Power hypertrees do not satisfy the reciprocal eigenvalue property.

Spectral symmetries are analyzed in the context of hypergraph structures.

Abstract

Spectral hypergraph theory has recently attracted considerable interest as it provides a natural framework for modeling higher-order relationships beyond classical graphs. In this setting, eigenvalues of adjacency, Laplacian, and signless-Laplacian hypermatrices play an important role in understanding the underlying structure of hypergraphs. In this work, we study the adjacency, Laplacian, and signless-Laplacian eigenvalues of the join, Kronecker, and corona products of hypergraphs, and examine how these spectra behave under such operations. These investigations help in better understanding the interplay between hypergraph structure and spectral properties. The reciprocal eigenvalue property is of particular interest due to the spectral symmetries it reflects. Motivated by this, we extend the notion of reciprocal eigenvalue property to hypergraphs and show that power hypertrees do not satisfy this property.