Algebraic Structure of Hypergraphs

Abstract

The aim of this work is to give a description on algebra associating to hypergraph, which is the generalized form of simple graphs and simplicial complexes. Algebraic study of combinatorial objects has been widely studied by different algebraist. Our main work in this regard is to associate an algebra to a special class of hypergraphs namely uniformly increasing hypergraphs H(X, E, d). This class of hypergraphs is im- portant in one sense that they can neither be regarded as simple graphs nor simplicial complexes. So, this work is the first stepping towards exploring the algebra associated to uniformly increasing hypergraphs H(X, E, d). In this thesis, we introduce inclusion ideals I(H) associated to the uniformly increasing hypergraphs H(X, E, d). We dis- cuss some algebraic properties of the inclusion ideals. In particular, we give an upper bound of the Castlenouvo-Mumford regularity of the special dual ideal I[ ](H).