Joins of Hypergraphs and Their Spectra

TL;DR

This paper explores the spectral properties of hypergraphs through matrix representations, extending equitable partitions and joins to compute eigenvalues of various hypergraph classes, including complete, loose cycles, and paths.

Contribution

It introduces new methods for spectral analysis of hypergraphs using matrix representations, extending equitable partitions and joins, and derives spectra for several hypergraph families.

Findings

Derived the characteristic polynomial of complete m-uniform m-partite hypergraphs.

Computed the spectrum of s-loose cycles for certain parameters.

Generated infinitely many pairs of non-isomorphic co-spectral hypergraphs.

Abstract

Here, we represent a general hypergraph by a matrix and study its spectrum. We extend the definition of equitable partition and joining operation for hypergraphs, and use those to compute eigenvalues of different hypergraphs. We derive the characteristics polynomial of a complete $m$-uniform $m$-partite hypergraph $K^m_{n_1,n_2,\dots,n_m}$. Studying edge corona of hypergraphs we find the complete spectrum of $s$-loose cycles $C^m_{L(s;n)}$ for $m \geq 2s+1$ and the characteristics polynomial of a $s$-loose paths $P^{(m)}_{L(s;n)}$. Some of the eigenvalues of $P^{(m)}_{L(s;n)}$ are also derived. Moreover, using vertex corona, we show how to generate infinitely many pairs of non-isomorphic co-spectral hypergraphs.