An Oriented Hypergraphic Approach to Algebraic Graph Theory

TL;DR

This paper introduces an oriented hypergraph framework with associated matrices, extending algebraic graph theory concepts to hypergraphs with signed incidences, and explores new matrix properties including walk matrices.

Contribution

It defines adjacency, incidence, and Laplacian matrices for oriented hypergraphs and extends known graph results, also presenting novel matrix results specific to hypergraphs.

Findings

Extended algebraic properties to oriented hypergraphs

Derived new matrix results not directly generalizable from graphs

Analyzed walk matrices within hypergraph context

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.