Abstract:
This dissertation addresses the algebraic invariants such as depth, Stanley depth,
projective dimension and regularity of some graphs. For this purpose, we investigate
certain types of monomial ideals of polynomial rings over the fields. Primarily, our
interest is to compute Stanley depth and depth of the edge ideals (or equivalently the
quotient of the polynomial ring by the monomial ideals) as module over the polynomial
ring. Indeed, this work is restricted to those ideals which are generated by square free
monomials of degree 2. The geometrical interpretation of such an ideal is the underlying
graph. This provides a bridge between algebraic objects and combinatorial objects,
that is, the monomial ideals and the graphs. From this correspondence, one can define
the algebraic invariants, Stanley depth and depth of graphs. The idea of projective
dimension is then explored through Auslander-Buchsbaum formula by using the former
concept of depth. Lastly, the regularity of underlying graph is computed by considering
minimal free resolutions of modules. In literature, some bounds and exact values of
depth and Stanley depth for the edge ideal associated with standard strong product
of graphs are given. This research is conducted to address such bounds of edge ideals
associated with restricted partial strong product, ladder and cubic circulant graphs.
Furthermore, we define a new family of circulant graphs and explore the algebraic
concepts of projective dimension and regularity by analyzing their bounds.
