Spectral Theory of Laplace Operators on Oriented Hypergraphs

Abstract

Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and used for proving a Hoffman-like bound. Finally, the spectrum of the unnormalized Laplacian for Cartesian products of hypergraphs is discussed.

arXiv comment39 pages

References in corpus (8)

• Lectures on 0/1-polytopes
• Coupled Dynamics on Hypergraphs: Master Stability of Steady States and Synchronization
• An Oriented Hypergraphic Approach to Algebraic Graph Theory
• Spectral Properties of Oriented Hypergraphs
• A Characterization of Oriented Hypergraphic Laplacian and Adjacency Matrix Coefficients
• Incidence Hypergraphs: Injectivity, Uniformity, and Matrix-tree Theorems
• Sharp bounds for the largest eigenvalue
• On the spectrum of the normalized graph Laplacian

Cited by in corpus (9)

• Structural reduction of chemical reaction networks based on topology
• $p$-Laplace Operators for Oriented Hypergraphs
• Coloring the normalized Laplacian for oriented hypergraphs
• Signless Normalized Laplacian for Hypergraphs
• A Cheeger Cut for Uniform Hypergraphs
• Measuring dynamical systems on directed hyper-graphs
• Geometry and symmetry in biochemical reaction systems
• Beyond directed hypergraphs: heterogeneous hypergraphs and spectral centralities
• Spectral theory of weighted hypergraphs via tensors