In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We als...

Let $G$ be a simple graph on $n$ vertices. $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal parity binomial edge of in polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] respectively. We classify graphs whose LSS ideals are complete intersections. also almost intersections, we prove that their Rees algebra is Cohen-Macaulay. compute second grade...

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. also provide examples oriented graphs that have non-Cohen-Macaulay vertex-weighted ideals, while the ideal their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes, Toledo.

We prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs.

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is c...