Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectra of Complex Unit Hypergraphs

Raffaella Mulas, Nathan Reff

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.

Spectra of Complex Unit Hypergraphs

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 or , as well as ordinary hypergraphs. Complex unit hypergraphs also generalize their graphic analogues, which are complex unit gain graphs, signed graphs, and ordinary graphs.

Paper Structure

This paper contains 10 sections, 21 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. First properties
  4. Hypergraph transformations
  5. Duality
  6. Weak vertex deletion and weak edge deletion
  7. Vertex and edge switching
  8. Smallest and largest eigenvalue
  9. Preliminary results
  10. Upper and lower bounds

Key Result

Theorem \oldthetheorem

The Kirchhoff Laplacian and the normalized Laplacian can be rewritten as respectively.

Figures (1)

  • Figure 1: A complex unit hypergraph $G$. Edge $e_1$ is a 4-edge, $e_2$ is an 2-edge, and edge $e_3$ has 3-edge. The incidence labels (incidence phase function values) are colored in blue. Here adjacency gain values for the two oriented adjacencies with $e_2$ are shown and colored in red. To make this picture much simpler, the other adjacency gain values are left out.

Theorems & Definitions (62)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 52 more