Date of Award
Doctor of Philosophy (PhD)
Dr. Michael Burr, Committee Chair
Dr. Elena Dimitrova
Dr. Shuhong Gao
Dr. Wayne Goddard
Dr. Svetlana Poznanović
Determining whether an arbitrary subring R of k[x1±1,...,xn±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 (xixj)±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 xi±1xj±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.
Lipman, Drew J., "Normal Domains Arising from Graph Theory" (2017). All Dissertations. 1920.