On the Regularity of Powers of Edge Ideals
Building on the work done by I. Swanson , S. Cutkosky, J. Herzog, and N. Trung published "Asymptotic behavior of the Castel Nuovo -Mumford regularity,"  which showed for sufficiently large powers s of an edge ideal I there exist positive values d, e such that the reg(J8) = ds + e where dis bounded above by the maximum degree of a generator of I. Since this discovery, work has been done to find a general form of the regularity for an edge ideal. This problem has been notoriously difficult to solve. As a result, most progress has been made by looking at the type of graph associated to the edge ideal. The paper "Regularity of Powers of Edge Ideals," by S. Beyarslan, H. T. Ha, and T. N. Trung  expands on the number of cases known by showing, for all powers of a forest and all powers greater than 1 of a cycle, reg(J8) = 2x·+ v(G) - 1 where v(G) is the induced matching number for the graph G. The purpose of this paper is to fill in the details of  and make the mathematics accessible to a graduate student. To facilitate this understanding, we include an extensive background section with fundamental definitions and useful lemmas and theorems. We also introduce new lemmas to the paper as well as provide alternate proofs where necessary to simplify the arguments made.
algebra, commutative algebra
Skelton, J. (2015). On the regularity of powers of edge ideals (Unpublished thesis). Texas State University, San Marcos, Texas.