Abstract. Let G be a finite simple graph on [n] and

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal I t (Γ ) is generated by monomials that correspond to directed paths of length (t − 1) in Γ . In this paper, we study algebraic properties and invariants of I t (Γ ). We give a recursive formula to compute the graded Betti numbers of I t (Γ ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when I t (Γ ) has a linear resolution.

Let G be a finite simple graph. We give a lower bound for the Castelnuovo-Mumford regularity of the toric ideal I G associated to G in terms of the sizes and number of induced complete bipartite graphs in G. When G is a chordal bipartite graph, we find an upper bound for the regularity of I G in terms of the size of the bipartition of G. We also give a new proof for the graded Betti numbers of the toric ideal associated to the complete bipartite graph K 2,n .

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