Algebraic view on neighborhood hypergraphstheir transversals, and d-partite hypergraphs

In this thesis, we explore the algebraic and homological properties of square-freemonomial ideals that originate from graphs and hypergraphs. Our study has twomain parts. In the first part, we study the closed neighborhood ideals and thedominating ideals of graphs. We prove that the closed neighborhood ideals and thedominating ideals of some classes of trees are normally torsion-free. However, theclosed neighborhood ideals and the dominating ideals of cycles fail to be normallytorsion-free. We prove that the closed neighborhood ideals of cycles admit the(strong) persistence property and the dominating ideals of cycles are nearly normallytorsion-free. Expanding our study to path graphs, we show the componentwiselinearity of dominating ideals of path graphs by describing a linear quotient order oftheir minimal generators. We also give formulas for their Betti numbers, regularity,and projective dimension.In the second part, we shift our focus to d-partite hypergraphs. Inspired by thedefinition of t-spread monomial ideals; we introduce the t-spread d-partite hypergraphs.The edge ideals of these hypergraphs, denoted by I(KtV ), admit some niceproperties. Namely, I(KtV ) has linear quotients and satisfies the ℓ-exchange propertyand the strong persistence property. Moreover, all powers of I(KtV ) have linearresolutions and the Rees algebra of I(KtV ) is a normal Cohen-Macaulay domain. Wealso prove that I(KtV ) is normally torsion-free and give a complete characterizationof Cohen-Macaulay S/I(KtV ).