Spectra of Complex Unit Hypergraphs

TL;DR

This paper introduces the concept of complex unit hypergraphs, defining their spectral properties and eigenvalue bounds, thereby generalizing various hypergraph and graph structures.

Contribution

It formalizes the spectral theory of complex unit hypergraphs and derives eigenvalue bounds, extending existing theories of hypergraphs and graphs.

Findings

Defined adjacency, incidence, Laplacian, and normalized Laplacian for complex unit hypergraphs

Established eigenvalue bounds for these matrices

Unified several hypergraph and graph structures under the complex unit hypergraph framework

Abstract

A complex unit hypergraph is a hypergraph where each vertex-edge incidence is given a complex unit label. We define the adjacency, incidence, Kirchoff Laplacian and normalized Laplacian of a complex unit hypergraph and study each of them. Eigenvalue bounds for the adjacency, Kirchoff Laplacian and normalized Laplacian are also found. Complex unit hypergraphs naturally generalize several hypergraphic structures such as oriented hypergraphs, where vertex-edge incidences are labelled as either $+1$ or $-1$, as well as ordinary hypergraphs. Complex unit hypergraphs also generalize their graphic analogues, which are complex unit gain graphs, signed graphs, and ordinary graphs.