Depth and Stanley Depth of Edge Ideals Associated with Quotient of Tadpole Graph and it’s Square Graph

Abstract

This dissertation aims to find a geometric invariant as well as the algebraic invariant of the graphs. The desire is to obtain required invariants such as depth and Stanley depth by using edge ideals as module over a polynomial ring as well as the quotient of polynomial ring by monomial ideals. For this purpose, square free monomial ideals are sought and critically reviewed. In this thesis, we find the exact values of depth and Stanley depth of quotient of the polynomial ring by the edge ideal associated with a Tadpole graph. It is shown that the values of these two invariants coincide. We also find the tight bounds for Sdepth and Depth of quotient of the polynomial ring by the edge ideal corresponding to square of a Tadpole graph.