Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all s ≥ 1, we obtain upper bounds for reg(I(G) s ) for bipartite graphs. We then compare the properties of G and G ′ , where G ′ is the graph associated with the polarization of the ideal (I(G) s+1 : e 1 · · · e s ), where e 1 , . . . e s are edges of G. Using these results, we explicitly compute reg(I(G) s ) for several subclasses of bipartite graphs.

Criteria are given in terms of certain Hilbert coefficients for the fiber cone F (I ) of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) so that it is Cohen-Macaulay or has depth at least dim(R) − 1. A version of Huneke's fundamental lemma is proved for fiber cones. Goto's results concerning Cohen-Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.

No abstract

We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid. We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of blockgraphs. As a consequence we obtain sharp upper and lower bounds for a class of trees called lobsters.

Abstract. Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.