We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We deter
mine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.
In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squar
efree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.
