Betti Numbers of Hypergraphs

Abstract

In this paper we study some algebraic properties of hypergraphs, in particular their Betti numbers. The Betti numbers of a hypergraph H are obtained from a minimal resolution of a certain quotient ring k[x1,..., xn]/I(H), the hypergraph algebra of H, where I(H) is a squarefree monomial ideal, the edge ideal of H. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. These can all be seen as natural generalizations of the ordinary complete graph Kn. As is often the case when investigating a hypergraph algebra, a great deal of attention is paid to its independence complex. Our main tools are Hochster’s formula, the Mayer-Vietoris sequence and Künneth’s tensor formula, all well known concepts in homological and combinatorial algebra. 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. 1