# A Characterization of Oriented Hypergraphic Laplacian and Adjacency Matrix Coefficients

## TL;DR

This paper extends graph theory concepts to oriented hypergraphs, providing a combinatorial characterization of their Laplacian and adjacency matrix coefficients, along with bounds on their determinants and permanents.

## Contribution

It introduces a signed hypergraphic generalization of basic figures to characterize matrix coefficients and analyzes bounds for hypergraph Laplacian matrices.

## Key findings

- Characterization of characteristic polynomial coefficients
- Bounds on Laplacian matrix determinant and permanent
- Identification of hypergraphs where bounds are sharp

## Abstract

An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved.