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