An Oriented Hypergraphic Approach to Algebraic Graph Theory

Abstract

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.

References in corpus (1)

• Matrices in the Theory of Signed Simple Graphs

Cited by in corpus (22)

• Spectral Properties of Oriented Hypergraphs
• Spectral Theory of Laplace Operators on Oriented Hypergraphs
• A Characterization of Oriented Hypergraphic Laplacian and Adjacency Matrix Coefficients
• Incidence Hypergraphs: Injectivity, Uniformity, and Matrix-tree Theorems
• The Signless Laplacian Matrix of Hypergraphs
• Normalized Laplace Operators for Hypergraphs with Real Coefficients
• Signless Normalized Laplacian for Hypergraphs
• Coloring the normalized Laplacian for oriented hypergraphs
• Sharp bounds for the largest eigenvalue
• A Cheeger Cut for Uniform Hypergraphs
• Oriented Hypergraphic Matrix-tree Type Theorems and Bidirected Minors via Boolean Order Ideals
• Geometry and symmetry in biochemical reaction systems
• Incidence Hypergraphs: Box Products & the Laplacian
• A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs
• Spectral classes of hypergraphs
• Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra
• The Determinant of $\{\pm 1\}$-Matrices and Oriented Hypergraphs
• Spectral theory of weighted hypergraphs via tensors
• Interlacing Results for Hypergraphs
• Generalizing Kirchhoff laws for Signed Graphs
• Laplacians on quantum hypergraphs
• Oriented Hypergraphs I: Introduction and Balance