On spectral hypergraph theory of the adjacency tensor

TL;DR

This paper explores the spectral properties of the adjacency tensor of uniform hypergraphs, focusing on eigenvalues, eigenvectors, and symmetry conditions, advancing hypergraph spectral theory.

Contribution

It provides new conditions linking eigenvalues to positive eigenvectors and characterizes symmetry in the E-spectrum of adjacency tensors.

Findings

Largest positive H or Z-eigenvalue has a positive eigenvector under certain conditions

Conditions for symmetry in the E-spectrum of the adjacency tensor

Insights into spectral properties of uniform hypergraph adjacency tensors

Abstract

We study both $H$ and $E/Z$-eigenvalues of the adjacency tensor of a uniform multi-hypergraph and give conditions for which the largest positive $H$ or $Z$-eigenvalue corresponds to a strictly positive eigenvector. We also investigate when the $E$-spectrum of the adjacency tensor is symmetric.