A general switching method for constructing E-cospectral hypergraphs

TL;DR

This paper introduces a general switching method for constructing hypergraphs with identical E-spectra, unifying and extending previous techniques in spectral hypergraph theory.

Contribution

It presents a comprehensive framework for generating E-cospectral hypergraphs through switching, generalizing existing methods and introducing new constructions.

Findings

The method unifies previous approaches to E-cospectral hypergraph construction.

It generalizes most existing switching tools for hypergraphs.

Standard methods for E-characteristic polynomial computation are often uninformative for hypergraphs.

Abstract

Spectral hypergraph theory studies the structural properties of a hypergraph that can be inferred from the eigenvalues and the eigenvectors of either matrices or tensors associated with it. In this paper we study the spectral indistinguishability in the hypergraph setting. We present a general switching method to construct uniform $E$-cospectral hypergraphs (hypergraphs with the same $E$-spectrum), and discuss some of its multiple applications. Our method not only provides a framework to unify the existing methods for obtaining $E$-cospectral hypergraphs via switching, but also generalizes most of the existing switching tools, yielding multiple new constructions. Finally, we compare common methods of computing $E$-characteristic polynomials, and in particular show that one standard method, while useful for generic tensors, is uninformative for almost all hypergraphs.