Let G be a finite simple graph and I(G) denote the corresponding edge ideal. In this paper we prove that if G is a unicyclic graph then for all s ≥ 1 the regularity of I(G) s is exactly 2s + reg(I(G)) − 2. We also give a combinatorial characterization of unicyclic graphs with regularity ν(G) + 1 and ν(G) + 2 where ν(G) denotes the induced matching number of G.
Abstract. In this paper we prove that if I(G) is a bipartite edge ideal with regularity three then for all s ≥ 2 the regularity of I(G) s is exactly 2s + 1.
We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are non zero and give a formula to compute its projective dimension and regularity.
Abstract. We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of paths and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of paths.
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