Abstract:
In this Thesis, we discuss rings and modules, which are the fundamental
algebraic structures of Abstract Algebra. Moreover, we discuss two algebraic
invariants of a module which are Stanley depth and depth of a module. In
addition, we discuss how the regular element property helps in determining
the depth of a module. Then, we discuss some recent results of depth and
Stanley depth of edge ideals associated with di erent graphs. Thereafter, we
nd the Stanley depth of some modules and edge ideals using the method
of posets. Lastly, we compute the Stanley depth and depth of the quotient
ring of edge ideals associated with di erent classes of graphs. These classes
include some lobster trees and unicyclic graphs. Then, we show that the
values of depth and Stanley depth are equal and can be stated in terms of n
and m. In addition, we prove the Stanley's inequality for modules associated
with these classes of graphs.
