Date of Award
5-2023
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
Mathematical Sciences
Committee Chair/Advisor
Keri Ann Sather-Wagstaff
Committee Member
Michael Burr
Committee Member
Beth Novick
Abstract
This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in [7] and by Herzog and Hibi in [6]. In 2020, Sharifan and Moradi [10] introduced a related construction called the closed neighborhood ideal of a graph. Whereas an edge ideal of a graph G is generated by monomials associated to each edge in G, the closed neighborhood ideal is generated by monomials associated to its closed neighborhoods. In 2021, Sather-Wagstaff and Honeycutt [8] characterized trees whose closed neighborhood ideals are Cohen-Macaulay. We will provide a generalization of this characterization to chordal graphs and bipartite graphs. Additionally, we will survey the behavior of the depth of closed neighborhood ideals under certain graph operations.
Recommended Citation
Leaman, Jackson, "Cohen-Macaulay Properties of Closed Neighborhood Ideals" (2023). All Theses. 3993.
https://tigerprints.clemson.edu/all_theses/3993