Date of Award
Master of Science (MS)
Keri Ann Sather-Wagstaff
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 : 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  and by Herzog and Hibi in . In 2020, Sharifan and Moradi  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  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.
Leaman, Jackson, "Cohen-Macaulay Properties of Closed Neighborhood Ideals" (2023). All Theses. 3993.