Abstract:

Let R = K[x1,x2,...,xn] be a polynomial ring in finitely many variables over a based field K. We consider modules over R. Such a module has a resolution and consequently a minimal resolution. Associated with the minimal resolution are some
integers known as the betti numbers. We will consider the edge ideal, which was first defined by Rafael H. Villareal, and the more general concept of secant powers of the edge ideal and derive a formula for computing the Betti numbers of edge ideals and secant powers of edge ideals of graphs obtained from complete graphs after the removal of some number of nonadjacent edges.