Algebraic Properties of the Path Ideal of a Tree
By Jinghua He, Adam Van Tuyl. 2009
73 citations recorded by the source corpus.
The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I t (Γ) when Γ is a tree. We first show that I t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.
View Paper: https://www.semanticscholar.org/paper/e45a78f2ef9940955c56d19cd8035adb4cb69fc9
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- Adam Van Tuyl
- Susan Morey
- Rafael H. Villarreal
- K. Kimura
- Huy Tài Hà
- Jürgen Herzog
- Arindam Banerjee
- Russ Woodroofe
Citations in the corpus
- Edge ideals: algebraic and combinatorial properties (2010, background): “1), (d) [55] clutters of paths of length t of directed rooted trees, (e) [39] simplicial forests, i.”
- A Beginner’s Guide to Edge and Cover Ideals (2013, background): “Further properties have been developed by He and Van Tuyl [35], Bouchat, Hà, and O’Keefe [3], and Alilooee and Faridi [1].”