Betti numbers of hypergraphs

TL;DR

This paper explores algebraic properties of hypergraphs, introduces new types of complete hypergraphs, and provides formulas for their Betti numbers, enhancing understanding of hypergraph algebraic invariants.

Contribution

It introduces new types of complete hypergraphs and a dual product operation, along with a general Betti number formula for these hypergraphs.

Findings

Defined new types of complete hypergraphs

Established a product operation dual to join on simplicial complexes

Derived a general Betti number formula for hypergraphs

Abstract

In this paper we study some algebraic properties of hypergraphs, in particula their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.