Spectral Theory of Hypergraphs: A Survey

TL;DR

This survey provides a comprehensive overview of the spectral theory of hypergraphs using hypermatrices, highlighting theoretical insights, computational methods, and diverse applications across multiple scientific domains.

Contribution

It offers an in-depth theoretical framework for hypergraph spectral analysis, connecting hypermatrices with spectral properties and recent advances in the field.

Findings

Overview of hypergraph spectral properties

Bounds for spectral radius of uniform hypergraphs

Relationships between characteristic and matching polynomials

Abstract

Hypergraphs require higher-dimensional representations, which makes it more difficult to compute and interpret their spectral properties. This survey article uses the framework of hypermatrices to give an in-depth overview of the spectral theory of hypergraphs. Our focus in this article relies on the theoretical aspects of hypergraphs that help to ease the computational methods. Spectral theory hypergraphs, one of the most advanced fields of study, are constantly finding novel applications in various domains such as theoretical computer science, quantum physics, and theoretical chemistry, among many others. We start our journey by introducing hypergraphs, hypermatrices (tensors), resultants, and their properties. We outline some of the results used to determine the adjacency spectrum of hypergraphs and go over some of the groundbreaking findings in the development of the theory. On passing through a list of bounds for the spectral radius of uniform hypergraphs, we will have a look into the spectral versions of Tur\'an-type problems in hypergraphs. Finally, in addition to the Estrada index of hypergraphs, some significant results related to the characteristic polynomial and its relationship with the matching polynomial are presented.