Some algebraic properties of hypergraphs

TL;DR

This paper explores algebraic properties of hypergraphs by analyzing their Stanley--Reisner rings, computing Betti numbers for certain classes, and extending results from graph theory to hypergraphs.

Contribution

It introduces new algebraic insights into hypergraphs, generalizes known graph results, and computes Betti numbers for specific hypergraph classes.

Findings

Betti numbers computed for hypergraph classes generalizing lines and cycles

Extended results on chordal graphs to hypergraphs

Studied weak shellability in hypergraph complexes

Abstract

We consider Stanley--Reisner rings $k[x_1,...,x_n]/I(\mc{H})$ where $I(\mc{H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize upon some known results about chordal graphs and study a weak form of shellability.