Depth and Stanley depth of the edge ideals of some classes of graphs

Abstract

In 1982, Stanley suggested the prominent conjecture in which he estimated a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The estimated invariant is now named as the Stanley depth. The Stanley conjecture is attractive in the sense that it compares a homological invariant with a combinatorial invariant of the module. In 2015, Duval et al. constructed a counterexample for Stanley’s conjecture. However, there still looks to be a profound and attractive relationship between these two invariants, which is yet to be understood. Furthermore, it is still fascinating to confirm Stanley’s inequality for some classes of modules and as result a lower bound for the Stanley depth can be achieved. The study of Stanley depth for modules is a complex problem. Herzog, Vladoiu and Zheng gave a combinatorial method to find Stanley depth. However, it is too difficult to calculate Stanley depth by their method because this is based on hard combinatorial techniques. The aim of this thesis is to provide the values and bounds of Stanley depth and depth of the edge ideals and quotient rings of the edge ideals associated with some classes of graphs. Furthermore, thesis gives a positive answer to Stanley’s inequality for quotient rings of the edge ideals related to some classes of graphs. In addition, a positive answer is also given to the Conjecture of Herzog for the edge ideals associated with some classes of graphs.