Algebraic properties of the path ideal of a tree

2009/02/05 by Jing Jane He, He, Jing Jane, Adam Van Tuyl +1 · 1 citation
Computer Science · Mathematics · Medicine · #05C99 #13D02 #13F55 #13P10 #Cholinesterase and Neurodegenerative Diseases #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0902.0902

openalex publication_date 2009/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebraic properties of I_t(G) when G is a tree. We first show that I_t(G) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I_t(G) is always sequentially Cohen-Macaulay, and the Betti numbers of R/I_t(G) 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 paper.

Citations

• The Betti numbers of forests

Cited by

• Path ideals of rooted trees and their graded Betti numbers

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