# On the Regularity of Powers of Edge Ideals

 dc.contributor.advisor Morey, Susan dc.contributor.author Skelton, Joseph dc.contributor.committeeMember Keller, Thomas dc.contributor.committeeMember Acosta, Teodora dc.contributor.committeeMember Oh, Suho dc.date.accessioned 2023-12-19T23:45:37Z dc.date.available 2023-12-19T23:45:37Z dc.date.issued 2015-05 dc.description.abstract Building on the work done by I. Swanson [18], S. Cutkosky, J. Herzog, and N. Trung published "Asymptotic behavior of the Castel Nuovo -Mumford regularity," [17] 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 [2] 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 [2] 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. dc.description.department Mathematics dc.format Text dc.format.extent 84 pages dc.format.medium 1 file (.pdf) dc.identifier.citation Skelton, J. (2015). On the regularity of powers of edge ideals (Unpublished thesis). Texas State University, San Marcos, Texas. dc.identifier.uri https://hdl.handle.net/10877/17802 dc.language.iso en dc.subject algebra dc.subject commutative algebra dc.title On the Regularity of Powers of Edge Ideals dc.type Thesis thesis.degree.department Mathematics thesis.degree.discipline Mathematics thesis.degree.grantor Texas State University thesis.degree.level Masters thesis.degree.name Master of Science

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