Betti Numbers of Hypergraphs

Abstract

In this paper we study some algebraic properties of hypergraphs, in particular 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.

A class of hypergraphs that generalizes chordal graphs

“…Note that the edges in H are precisely the minimal non faces in (H ). The connections between a (hyper)graph and its independence complex are explored in, for example [5], [8], [14]. In Section 2 we will see that in case of simple uniform hypergraphs H , there is a very natural connection between the independence complex of H and the complex of H .…”

“…After that, hypergraph algebras have been widely studied. See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19]. In [10], the authors use certain connectedness properties to determine a class of hypergraphs such that the hypergraph algebras have linear resolutions.…”

“…In [5], we considered hypergraph generalizations of the well known complete and complete multipartite graphs. We use these to create some examples of chordal hypergraphs.…”

A Beginner’s Guide to Edge and Cover Ideals

“…(Again, I'm glossing over some details about the isolated vertices.) Properties of these ideals have been studied by Emtander [19], Francisco, Hà, and Van Tuyl [25], Hà and Van Tuyl [32] and Morey, Reyes, and Villarreal [51], among others. These ideals have also been called facet ideals; see, for example, Faridi [21].…”

Betti numbers of mixed product ideals

“…After this paper has been submitted the author learned that there are slight overlappings with the recent paper (see [3]). In particular Theorem 3.1 and Corollary 3.2 of [3] are related with our result in Lemma 3.1.…”