Algebraic Properties of the Path Ideal of a Tree

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