Spectral Theory of Hypergraphs: A Survey

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.