A V Jayanthan

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of I(G)q in terms of certain combinatorial invariants associated with G. We also prove a weaker version of a conjecture by Alilooee, Banerjee, Beyarslan and Hà on an upper bound for the regularity of I(G)q and we prove the conjectured upper bound for the class of vertex decomposable graphs. Using these results, we explicitly compute the regularity of I(G)q for several classes of graphs.

Let G be a finite simple graph on n vertices and JG denote the corresponding binomial edge ideal in S = K[x1,…, xn, y1,…, yn] We compute the Castelnuovo-Mumford regularity of S/JG when JG is the binomial edge ideal of a Cohen-Macaulay bipartite graph. We achieve this by computing the regularity of certain bipartite subgraphs and some intermediate graphs, called k-fan graphs. In this process, we also obtain a class of graphs which satisfy the regularity conjecture of Saeedi Madani and Kiani.

Let G be a finite simple graph on n vertices. Let JG ⊂ K[x1,..., xn] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo-Mumford regularity of JsG for all s ≥ 1 for certain classes of graphs G.

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of JGs is bounded below by 2s+ℓ(G)−1 for all s≥1, where JG denotes the binomial edge ideal of a graph G and ℓ(G) is the length of a longest induced path in G. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.

Let G be a finite simple graph and I (G) denote the corresponding edge ideal. We prove that if G is a very well-covered graph then for all s > 1 the regularity of I(G)s is exactly 2s + v(G) - 1, where v(G) denotes the induced matching number of G.

In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.

In this article, we study the Castelnuovo-Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone. © 2012 Copyright Taylor and Francis Group, LLC.

In this article, we obtain an improved upper bound for the regularity of binomial edge ideals of trees.