Depth, Stanley Depth and Regularity of Edge Ideals Associated with Some Graphs

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.