Date of Award


Document Type


Degree Name

Master of Science (MS)


Mathematical Sciences

Committee Chair/Advisor

Keri Ann Sather-Wagstaff

Committee Member

Michael Burr

Committee Member

Beth Novick


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.



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