Normal Domains Arising from Graph Theory

Author

Drew J. Lipman, Clemson UniversityFollow

Date of Award

5-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematics

Committee Member

Dr. Michael Burr, Committee Chair

Committee Member

Dr. Elena Dimitrova

Committee Member

Dr. Shuhong Gao

Committee Member

Dr. Wayne Goddard

Committee Member

Dr. Svetlana Poznanović

Abstract

Determining whether an arbitrary subring R of k[x₁^±1,...,x_n^±1] is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. First, we determine normality in the case where R is a monomial generated domain where the generators have the form (x_ix_j)^±1. Using results for this special case we generalize to the case when R is a monomial generated domain where the generators have the form x_i^±1x_j^±1. In both cases, for the ring R, we consider the combinatorial structure that assigns an edge in a mixed directed signed graph to each monomial of the ring. We then use this relationship to provide a combinatorial characterization of the normality of R, and, when R is not normal, we use the combinatorial characterization to compute the normalization of R. Using this construction, we also determine when the ring R satisfies Serre's R₁ condition. We also discuss generalizations of this to directed graphs with a homogenizing variable and a special class of hypergraphs.

Recommended Citation

Lipman, Drew J., "Normal Domains Arising from Graph Theory" (2017). All Dissertations. 1920.
https://open.clemson.edu/all_dissertations/1920